Excited states of carbonyl compounds. 1. Formaldehyde and

J . Phys. Chem. 1993,97, 4293-4312

4293

Excited States of Carbonyl Compounds. 1. Formaldehyde and Acetaldehyde Christopher M. Hadad, James B. Foresman, and Kenneth B. Wiberg’ Department of Chemistry, Yale University, New Haven, Connecticut 0651 1 Received: September 8, 1992; In Final Form: December 21, 1992 The electronically excited states of formaldehyde and acetaldehyde are examined with the configuration interaction with all single excitations (CIS) method. Vertical and adiabatic transition energies for singlet and triplet states are calculated, and generally good agreement with experiment is obtained for valence states a t the CIS level and for both valence and Rydberg states at the CIS-MP2 level. Analysis of the charge density distribution, via difference plots with respect to the ground-state distribution, clearly revealed the nature of the excitation and allowed for an unambiguous assignment of the valence and atomic-orbital-like Rydberg states. The adiabatic geometries of the singlet and triplet states are calculated, and they agree well with experiment. The derived vibrational frequencies yielded good agreement with experiment for both valence and Rydberg excited states. Information is provided on the experimentally unassigned a a* transitions in both carbonyl compounds. The charge density analysis provided quantitative values for the amount of charge transfer in the excited states and also revealed the preferences for the different excited states in their adiabatic geometries. The geometrical distortions and the underlying charge transfer are discussed.

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Introduction Carbonyl compoundsplay an important role in chemistry from physical-organic to photochemical, synthetic, and biological studies.’ Many of these compounds have been investigated to study the reactivity of the carbonyl group and its preferences for substituents in the ground state.2 There have been fewer studies of the excited states of these compounds, yet the excited states are quite interesting due to the presence of both a a system and the lone pairs on oxygene3 As a result both n a* and a a* transitions are known for carbonyl compound^,^ and they are responsible for important photochemical transformations. In particular, Norrish types I and I1 reactions have been extensively studied and utilized in organic chemistry.5 Of the carbonyl compounds, the smaller systems such as formaldehyde, acetaldehyde, and acetone have been investigated by a variety of experimental4 and theoretical6 methods. Many vertical excitations have been assigned, and some information is available on the geometries of the excited states as well as some of their vibrational frequencies. There have also been many experimental studies on the dissociation pathways for ketones in the excited states,’ and these states have proven useful as a source of free radicals. Few theoretical studies of the dissociation pathways have been undertaken8 While theoretical methods have been applied to the study of the ground states of organic molecules, excited states havereceived much less study. However, the recent implementation of the configuration interaction with all single excitations (CIS) method by Foresman et ala9has provided a useful method for studying both valence and Rydberg excited states. The CIS method has been applied to the excited states of bicyclo[ l.l.0]butane,I0 ethylene,” and 1,3-butadienei2with considerable success. Formaldehyde has previously been studied by the CIS method, and to extend this study, we examined the excited states of acetaldehyde and acetone. In this paper we present a comprehensive study of the singlet and triplet states of formaldehyde and acetaldehyde. Our analysis of the excited states of acetone will be presented subsequently. For these compounds, we can explore the equilibrium geometries of theexcitedstates and obtain thecorrespondingvibrational frequencies. These data will allow some comparison of the theoretical results with the experimental facts and, thus, will provide a good test of the CIS theoretical method for the calculation of excited-stateproperties. Moreover,one of our major

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interests is to explore the relative charge distributions of the different excited states, especially the n a* and a a* states, and to see how they differ from each other and from the Rydberg states of carbonyl compounds. The charge distribution may also provide some insight into the origin of the interesting photochemical behavior which is observed for carbonyl compounds. The vertical n a* transition energy is known to increase with increasing methylation from formaldehyde (4.07 eV) to acetaldehyde (4.28 eV) to acetone (4.43 eV).4 One explanation is as follows. From isodesmic reactions, a carbonyl group is stabilized by -7 kcal mol-] when the substituent H is replaced by a methyl group.I4 The change in then a*transition energy with increasing methylation suggests that some of the methyl stabilization for the ground state is lost upon going to the n T * excited state. We will try to address this hypothesis with our analysis of the charge density distributions. We have therefore undertaken a comprehensive study of the excited states of formaldehyde and acetaldehyde. We have obtained information on the vertical and adiabatic transition energies for the singlet and triplet states of these compounds as well as thegeometricdistortions that occur in thedifferent excited states. We will also investigate the methyl effect on valence excitations.

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Formaldebyde As the simplest carbonylcompound,formaldehydehas received much experimentaland theoretical study. The experimentalwork to 1975 was reviewed by Moule and Walsh15 and again in 1985 by Robin. Davidson and McMurchie have reviewed the theoretical work on formaldehyde.16 While agreement on thevertical transition energies has been quite good, there has been much less information about the geometries of the excited states and the corresponding vibrational frequencies. The lowest singlet state corresponds to the n, a*transition, and this state has been well characterized.” It is known to be bent in the excited state, and the vibrational frequenciesare known. Less is known about the Rydberg states. The information on the n, a* state will allow a test of our theoretical calculations.

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Vertical Singlet Transition Energies Our previous studies1”I2 with the CIS method have shown that diffuse functionsarevery important for an adequate treatment of the excited states of organic compounds. We have therefore examined the sensitivity of the vertical transition energies to the

0022-365419312097-4293%04.00/0 0 1993 American Chemical Society

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The Journal of Physical Chemistry, Vol. 97, No. 17, 1993

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TABLE I: Calculated Vertical Singlet Transition Energies for Formaldehyde (eV)a 6-31 1+G* 1 2 3 4 5 6 7 8 9 10 11

state

CIS

1A2 1B2 2AI 2B2 1BI 3Al 2A2 3A2 2Bl 3Bl 4.42

4.51 8.90 9.56 9.57 9.69 10.03 10.41 11.23 11.31 12.12 12.30

12 13 14 15 16

3B2

17 18 19 20

6-31 1(2+)G* oscstr

state

CIS

0 0.0230 0.3014 0.0559 0.0005 0.0053 0 0 0.1044 0.0152 0

1A2 1B2 2B2 2A1 1B1 3A1 2A2 3B2 3A2 2B1 4B2

4.51 8.64 9.38 9.48 9.69 9.70 9.78 10.68 10.94 11.06 11.08

AI

12.78 12.85 13.24 14.21 14.23

0.2866 0.0502 0.0011 0 0.1790

6Al 4Bl 7.41 5B2

14.73 14.99 15.09 15.12

0.0033 0.0243 0.2751 0.0015

AI 4B2 5A2

6-31 1(2+,2+)G** oscstr

state

0 0.0215 0.0426 0.2541 0.0005 0.0359 0 0.0135 0 0.0724 0.0030

1A2 1B2 2B2 2A) 3AI 1BI 2-42

3B2 4B2

AI 3.42

4A2 5A1 ~ B I

11.15 11.17 11.40 11.70 11.85

0.0324 0.0005 0 0.0718 0.0200

5B2 2B1 6B2 5A1 4A2

0 0 0.0109 0.2077

3B1

6A2 6Ai 6B2

11.94 12.07 12.17 13.03

5B2

AI

6A2 6Al

assignment n,-r* n,-3s ny- 3p, r+r*

ny-3py n,-r* n,-3pX n,- 3d,2 n, 3d,2 ny-4d,,

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+ 4s

-

TCH,-T*

n, 3dx, n,- 3dX?-p r - 3 ~ n, 4p,, 4d9 nv-4p, R C H ? - ~ ~ ~

ny 4dx2 r-3p2 T43py n,-4p, ny-4fyr?

CIS

oscstr

CIS-MP2

exptlb

4.48 8.63 9.36 9.45 9.66 9.66 9.78 10.61 10.86 10.88 10.92

0 0.0243 0.0468 0.2608 0.0148 0.0004 0 0.0154 0.0025 0.0059 0

4.58 6.85 7.66 9.19 8.47 9.97 7.83 8.46 8.94 8.75 10.08

4.07 7.1 1 7.97

10.98 11.05 11.17 11.24 ~ 11.37

0.0761 0.0707 0.0025 0.0967 0

8.96 10.84 9.19 9.20 10.49

11.84 11.93 12.06 12.09

0.0217 0 0 0.0082

11.56 11.63 10.13 9.99

8.14 8.37‘ 8.88 9.22c 10.60

11.70

Using the MP2/6-31G* ground-stategeometry. Theground-stateenergies at the RHF/6-311+G*, RHF/6-311(2+)G*, RHF/6-311(2+,2+)GS*, and MPZ(ful1)/6-311(2+,2+)G** levels are -1 13.896 43, -1 13.896 996, -1 13.900 047, and -1 14.287 066 hartrees, respectively. Reference 4, p 256. Reference 30. (I

choice of basis set. Using the MP2/6-31G* geometry for the ground state (see below), which is similar to the experimental geometry,I8 we have calculated the vertical singlet transition energies with the 6-31 l+G*, 6-31 1(2+)G*, and 6-311(2+,2+)G** basis sets where the latter basis set, for example, is the 6-31l++G** basis set with an additional diffuse function on all atoms that has an exponent which is smaller by a factor of 3.32 from the first diffuse function. The vertical transition energies are listed in Table I. First of all, one sees that the high-energy states require a second diffuse function on heavy atoms for a reasonable treatment. The diffuse functions on hydrogen, while balancing the basis set, only affect the transition energies to a small degree, decreasing them by -0.1 eV on average. The low-energy states are less affected by extra diffuse functions. Three states, in particular, 1A2,2A1, and lBl, are relatively unaffected by the basis set flexibility, and they correspond to valence states (see below). Before we compare the calculated and experimental vertical transition energies, we explain our method of assignment for the calculated excited states. While previous researchers have assigned their theoretical states via just the symmetry of the final states,6we have instead analyzed the wave functionsfor the excited statesat the CIS level. We have thereforeobtained thegeneralized density matrix for each excited state,9 and this is transformed into natural orbitals, thereby allowing the charge density to be determined as a function of the coordinate. By using charge density difference plots for the excited states as compared to the ground state, it is relatively straightforward to assign the nature of the excited-state transition. In practice, we compute a cube of charge density (30 au on each side and 80 points per side) for both the excited state and the ground state. Subtraction of the ground-state distribution from that of the excited state shows how the charge density is shifted upon electronic excitation. Visualization of the 3-D density difference plots is performed via a modification of the program described by Jorgensen.19 The density difference plots for the singlet excited states of formaldehyde are shown in Figure 1. Positive (solid) contourscorrespond to an accumulation of charge density in the excited state relative to the ground state, and negative (dashed) contours correspond to charge depletion. In Figure 1, we have shown our theoretical assignment for the singlet excited states, and one can see transitions which are atomic-

orbital-like Rydberg states as well as valence states. Many of the states originate from the highest occupied molecular orbital (n,), but some of the states arise from the x and n, molecular orbitals. Plots of t/t for the four highest occupied molecular orbitals of formaldehyde are shown in Figure 2. We should note that the n, occupied molecular orbital (MO) is not localized on oxygen and, therefore, is not simply a “nonbonding” MO, but rather, the n, MO is delocalizedover the entire molecule. This is in agreement with previous suggestions.6BJO The charge density difference plots in Figure 1 show that the T * valence states are significantly more compact than the Rydberg states. This compactnessis the reason for the relative insensitivity ofthevalencestatestodiffuse functionsin thebasisset. (However, one diffuse function was needed.) One can see many s, p, and d Rydberg states that arise from the n, MO, and s and p Rydberg states from the x MO. It is interesting to note the 2AI and 3AI states. Their charge density difference plots appear to be quite similar; however, each is composed of a mixture of x x* and n, 3p, configurations. Both configurations have the same overall symmetry (AI), but the originating MO differs in symmetry (n,is b ~ a n d x ibl). s Theseconfigurationscan therefore mix, and in the 2AI state, the T T * contribution is major and the n, 3p, transition is minor. The 3AI state is the opposite and has the n, 3p, contribution being larger. (We should note that therelativemixingof these twoconfigurationswill be sensitive to both the energy separation and the geometry of the molecule as discussed below.) An easier method of differentiating the Rydberg and valence states is to examine just the depletion regions (dashed contours) for theverticalexcited states. Figure 3 shows thedepletion regions for the first eight singlet excited states at the CIS level and the vertical n, radical cation at the UHF/6-3 1 1(2+,2+)G** level. One can see the similarity of the n, Rydberg states to the n, radical cation and, moreover, the difference from the valence states. Even the mixedvalencestates, 2A1 and 3A1, havedepletion regions that are different from that of the radical cation. The 2B2 state also shows a charge depletion region that is different from that of the radical cation, and this difference is probably due to a slight contribution of the T C H ~ x* configuration. In Table I we have listed the known experimental vertical transition energies (or adiabatic, if vertical is not known) for the singlet excited states of f~rmaldehyde.~ Our charge density

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Excited States of Carbonyl Compounds

The Journal of Physical Chemistry, Vol. 97, No. 17, 1993 4295

3 p z 28: ( 3 )

ny

ny

- B?,

212 ( 7 )

:ldz2

ny

t

fi

4 s 482 ( 9 )

I b l (71

n, l'bz (8)

Figure 2. )I for the four highest occupied molecular orbitals of formaldehyde at the 6-311(2+,2+)G** level (0.05 au for the outer contour).

ny

-

3 d , 2 - r 2 582 ( 1 2 )

I J

Figure 1. Charge density difference plots for the first 18 vertical singlet excited states of formaldehyde at the CIS/6-311(2+,2+)G** level with respect to the ground state. The outermost contour is 0.0001 e/B3, except for state 16 which is 0.000 05 e/B3.

difference plots allow for a clear and reliable assignment of the transitions, and we obtain the correct ordering of the valence and Rydberg singlet states at the CIS/6-3 11(2+,2+)G** level. The CIS method provides good agreement for the n,, R* state, whereas the CIS energies for the Rydberg states are generally high by -1.5 eV. It is also possible to include a correction for electron correlation to the CIS energiesvia Maller-Plesset perturbation theory through second order (CIS-MP2).9.21Our previous studies1c12haveshown that the CIS-MP2 energies provide better agreement with experimental transition energies for systems in which the ground state is not described well by Hartree-Fock theory. Thus, while CIS-MP2improvesthe agreement for the excited states of bicycle[ 1.1.O] butane,IO the CIS method provides better agreement for ethylene' I and 1,3-butadiene.12 We also calculated the CISMP2 energies for formaldehyde with the6-311(2+,2+)G** basis set. The valence states are affected only slightly by the CISMP2 correction, and in general, the excitation energies increase when compared to CIS. The Rydberg states, on the other hand,

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+

3 ~ 2: 1 2 ( 7 )

07

3dz2 382 (8)

n, Radical C a t i o n 2 8 2

Figure 3. Charge density depletion regions (negative contours) for the first eight vertical singlet excited states and the n,, radical cation of formaldehyde (0.0001 e/B3).

are affected more strongly, and the excitation energies decrease by -2 eV. The agreement with the experimental energies is improved at the CIS-MP2 level; typically the calculated values are within 0.5 eV. This effect was also observed by Foresman et al.9 We should note that good agreement is achieved for both n,, and ?r excited states. We have also calculated the vertical ionization potentials (IP) for formaldehyde using the MP2/6-3 1G*geometry of the ground state with the 6-31 1(2+,2+)G** basis set. The data are listed in Table 11. The n,,vertical experimentalIP2*is well reproduced at any level when electron correlation is included (even only at the MP2 level). The ?r vertical IP is also well reproduced at any level that includes electron correlation. The 6-3 1 1(2+,2+)G** basis set seems to be quite successful in describing the radical cations and the excited states. In Table I11 we have listed some of the calculated vertical transition energies that have been published to date6a*JJ*qJ6as

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Hadad et al.


TABLE II: Vertical Ionization Potentials. Formaldehyde (a) Absolute Energies (6-31 1(2+,2+)G**) HF 'AI GS 2B2 n, RC 2Bi T R C

-1 13.900 05 -113.552 75 -113.449 59

PUHF

MP2

-113.559 43 -113.451 43

-1 14.247 30 -1 13.838 05 -113.710 52

PMPZ

MP3

-113.843 34 -113.711 42

-1 14.249 62 -113.853 70 -113.732 36

PMP3

MP4

PMP4

-113.857 68 -113.73282

-1 14.275 02 -113.872 97 -113.74343

-1 13.876 95 -113.743 89

(b) Vertical Ionization Potential 2B2n, RC 2 B T~ RC

HF

PUHF

MP2

PMPZ

MP3

PMP3

MP4

PMP4

exptlb

9.45 12.26

9.27 12.21

11.14 14.61

10.99 14.58

10.77 14.08

10.67 14.06

10.94 14.47

10.83 14.45

10.88 14.39

Acetaldehyde (a) Absolute Energies (6-31 1(2+)G*) HF 'A'GS 2A' n RC

-152.953 48 -152.627 94

PUHF

MP2

-152.633 72

-153.428 23 -153.043 71

PMPZ

MP3

-153.048 08

-153.438 68 -153.068 30

PMP3

MP4

PMP4

-153.071 45

-153.471 50 -153.094 77

-153.097 97

(b) Vertical Ionization Potential

2A' n RC

HF

PUHF

MP2

PMPZ

MP3

PMP3

MP4

PMP4

exptlC

8.86

8.70

10.46

10.34

10.08

9.99

10.25

10.16

10.21

All MP calculations utilized the frozen-core approximation, and MP4 refers to MP4(SDTQ). The ground-state (GS) and radical cation (RC) energies are R H F and UHF, respectively. All projected MP energies correspond to annihilation of only the first spin contaminant. b References 22 and 25. As cited in ref 43.

TABLE III: Comparison of Vertical Calculated Transition Energies for Formaldehyde state

HGb

YMe

4.09 7.16 8.08 10.77 8.09

4.04 7.28 8.12 10.10 8.15 9.19 8.35

8.32 9.05 9.24 10.73 11.66 11.78 3.68 5.95 7.08 7.99 8.05 8.31

WHd GP' (a) Singlet States 3.80 3.67 7.48 7.15 8.15 11.31 10.43 8.30 7.97 9.35 8.85 8.12 11.23

3.46 5.29

PBKW

Gg

CIS-MP2h

exptl'

3.81 7.38 8.39 11.41 8.11

3.18 7.39 8.22 9.00 8.32 7.88 8.53 9.24 9.59

4.58 6.85 7.66 9.19 8.47 9.97 7.83 8.46 10.08 10.84 11.56 11.63

4.07 7.1 1 7.97

9.07

11.08 12.06 12.11

(b) Triplet States 3.38 3.29 5.66 6.16 6.76 7.66 7.51 8.10 7.73 8.12

3.41 5.56 7.32

4.15 6.72 6.97 9.18 7.75 7.78 8.16 10.52 8.67 9.12

8.29 8.09 9.06

8.14 8.37J 8.88 9.22 10.60 11.70

3.50 5.86 6.83 7.79 7.96

9.59

In unitsof electronvolts. Reference61. Reference6c. Reference6a. e Reference 6j. /Reference 6b. g Reference6q. The RPA excitation energies were determined with respect to the experimental ionization potential. This work, using the 6-31 1(2+,2+)G** basis set. Reference 4. j Reference 30.

well as the CIS-MP2/6-311(2+,2+)G** values and the experimental data for f~rmaldehyde.~ One can see generally consistent agreement between the CIS-MP2 energies and those from the differ from the previous methods, and the experimental transition energies by less than 0.4 eV for both valence and Rydberg states of the n and ?r excited states. In addition, the CIS and CIS-MP2 energies are determined via an absolute energy difference between the calculated ground and excited states and not as term values with respect to the experimentalverticalionizationpotential.@ Thus, theCIS method seems to be a viable technique for obtaining satisfactory transition energies of ketones, especially when experimental data are not available.

Adiabatic Singlet Excited States The adiabatic singlet states of formaldehyde have aroused a great deal of i ~ ~ t e r e s t .The ~ ~ ,presence '~ of vibrational structure in a few of the excited states has allowed experimental deductions on the bent nature of the n, ?r+ state. We have calculated the adiabatic geometries of a few of the excited states at the CIS/ 6-31 1 + ~ level, * and the geometrical parameters are listed in Table IV, The adiabatic geometry of the 2Bz(ny) radical cation has also been examined at the UHF/6-311+G* level. We shall discuss each state in turn. The n, ?r* state is known to be bent (C,) in the excited state with a r(H-C-0-H) angle of 1 4 8 O (32' bent from planar).z3 The Cz, geometry for the n, ?r* (1Az) state was calculated to

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TABLE IW Geometrical Parameters for Formaldehyde Ground and Singlet Excited States. (a) Ground State (C2J parameter

MP2/6-31G*

exptlb

RHF/6-311+G1

r(C-0) r(C-H) (H-C-H) (H-C-0) energyC

1.2200 1.1039 115.62 122.19

1.2078 1.1161 116.50 121.75

1.1788 1.0935 116.12 121.94 0.900 07

parameter 4C-O) r(C-H) L(H-C-H) L(H-C-0) r(H-C-O-H) NIMAGP vp

energyc CIS transn energf CIS-MP2 transn energy/g

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(b) Excited States (6-31 l+G*)

-.

ny- 3s 1B2 (C2J

ny- 3p,

1A2 (C2J

us 1A" (C,) 1A" (exptld)

2B2 (C2J

2A1 ( G o )

n, u* 1BI (Go) 1A' (C,)

1.2483 1.0836 121.04 119.48 180 1 403.33 0.732 91 4.55 4.35

1.2482 1.0877 117.25 117.72 149.35 0 529.5 0.733 41 4.54 4.33

1.1235 1.1736 100.18 129.91 180 0 762.5 0.587 3 1 8.51 7.70

1.1346 1.1460 103.77 128.12 180 0 494.0 0.559 34 9.27 8.33

1.4540 1.0741 124.38 117.81 180 0 934.7 0.582 09 8.65 8.60

1.4803 1.0685 142.60 108.70 180 1 1025.li 0.593 39 8.35 8.79

ny

1.321 1.097 118.6 148.25

3.50

A-u*

1.4886 1.0785 121.93 107.96 133.55 0 878.0 0.600 89 8.14 8.49

RC (UHF) 2B2 (Go) 1.2031 1.0900 124.04 117.98 180 0 1074.8 0.551 89 9.49 10.97

+

All bond distances in angstroms and angles in degrees. Reference 18. Energy is -(113 value) hartrees. Reference 23. e For this and all subsequent tables, NIMAG is the number of imaginary vibrational frequencies, and v corresponds to the lowest real or all imaginary frequencies (cm-I). (Transition energy in electronvolts. g The MP2(fu11)/6-31 l+G*//RHF/6-311+G* energy for the ground state is -1 14.270 25 hartrees.

have a slightly elongated C-0 bond length and an increased H-C-H bond angle. This structure is a transition state, and the vibrational mode corresponding to the imaginary frequency was the out-of-plane bending mode (4. Optimization under C, symmetry decreased the H-C-H angle to a value similar to that of the ground state, and with a dihedral angle T of 149', it is in very good agreement with the experimental value. The C-0 bond length, though, is significantlyshorter than the experimental value. Foresman et al. have shown that a further inclusion of electron correlation (CIS-MP2) for the geometry optimization causes an increase in the C-0 bond length.9 Therefore, the underestimation of the C-O bond length at the CIS level is similar to the known deficiency of Hartree-Fock theory in predicting C-O bond lengths to be too short, while MP2 geometries predict values closer to experimental values.24 Our calculated CIS adiabatic transition energy is 4.54 eV (3.50 eV, e~perimental),~~ andasinglepoint energycalculationat theCIS-MP2/6-311+G* level (using the CIS/6-3 lG* geometry) gave a transition energy of 4.33 eV. Thus, as seen with the vertical valence states, the adiabatic transition energy is not strongly affected by the MP2 correction. We have calculated the adiabatic geometries of the n, 3s (1B2) and ny 3p, (2B2) states. Both states show similar behavior. The C-O bond length is decreased by 0.1 A from the ground state, and the bond angles aredrastically affected. Unlike the ny A* state, the H-C-H angle is decreased to loo', and the H-C-0 angle is opened to 130'. Both C2" structures are minima. Similarly to the CIS vertical energies, the adiabatic transition energies are overestimated by 1.5 eV with respect to experiment: 8.51 and 9.27 eV (calculated) and 7.08 and 7.97 eV (e~perimental)]~ for the 1B2 and 2B2 states, respectively. However, the CIS-MP2 correction is quite important for these states (- 1 eV), and the calculated adiabatic transition energies at the CIS-MP2/6-311+G1 level are 7.70 and 8.33 eV, respectively. These values are in much better agreement with the experimental values. The ny radical cation (2B2) optimized geometry is also a minimum under C2"symmetry, but the C-O bond length is 0.07 A longer than that in the Rydberg states. Then, radical cation's H-C-H and H-C-0 angles increase and decrease, respectively, ascompared tothegroundstate. ThisisincontrasttotheRydberg excited states. The adiabatic IP for the radical cation is calculated to be too low at the UHF level (8.49 eV); however, a single point energy calculation at the UMP2/6-3 11+G* level using the UHF

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geometry provided an adiabatic IP of 10.97 eV, which is in good agreement with the experimental value (10.88 eV).22325 We have also examined the adiabatic geometries of the A A* (2A1) and n, A* (1BI) states. The A A* state has an elongated C-0 bond length (1.454 A), and the H-C-H angle has increased to 124'. Walsh has predicted that the A A* state could be bent,17band previous calculations have predicted that the state is dissociative.26 However, our calculations show that the structure is a minimum under CzVsymmetry. The C-O bond, though, is lengthened significantly. The adiabatic transition energy is calculated to be 8.65 and 8.60 eV at the CIS and CISMP2 levels, respectively. Under C2"symmetry the n, A* state similarly lengthens the C-0 bond (1.48 A) but drastically alters the H-C-H angle to 143'. As expected, the C2" geometry is a transition state, and the deformation corresponding to the imaginary frequency is an out-of-plane bend. The C, structure is a true minimum, and the H-C-H angle returns to 122', along with a dihedral angle ( 7 ) of 134O (46' bent fromplanar). Weobtain an adiabatic transition energyof8.49eVat theCIS-MP2/6-31 l+G*//CIS/6-311+G* level for this state. Some vibrational frequencies are also known for some of the excited states of formaldehyde.15 In particular, the n, A* state has been extensively studied.2' In Table V we have listed the calculated and experimental vibrational frequencies for the ground and excited states of formaldehyde. The experimental frequencies for the ground state have been well characterized.28 The RHF/6-311+G* vibrational frequencies for this state are too high by 10%as the bond lengths tend to be too short at the Hartree-Fock Typically, Hartree-Fock-derived vibrational frequencies can be scaled by -0.9 in order to reproduce experimental f r e q ~ e n c i e s .The ~ ~ scaling factors for the ground state are shown in Table V, and they range from 0.875 to 0.908. The excited-state vibrational frequencies calculated at the CIS/ 6-3 1+G* level are similar to those of the ground state, so we have utilized the ground-state scaling factors to scale the calculated CIS frequencies. This method has worked quite well for our previous calculations on bicyclo[ 1.1.O]butane,I0 ethylene," and 1,3-butadiene. I By visual inspection of the displacement vectors for the vibrational modes, we were able to correlate the calculated Ch and C, frequencies of the excited states to the Cz, modes for the ground state, and these are listed in Table V. Our scaled C, vibrational frequencies for the n, A* state agree quite well

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4298 The Journal of Physical Chemistry, Vol. 97, No. 17, 1993

Vibrational Frequencies' of Formaldehyde Ground and Singlet Excited States (6-311+G*)

TABLE V

ny

GS (1'AI) mode

assignment

VI

symC-Hstretch C-0 stretch CHI scissor out-of-planebend asymC-Hstretch CHzrOCk

AI

v2 v3 v4

BI B2

v5 v6

-

mode AI

VI

v2 v3

BI

v4

B2

vs v6

RHF 3126 1996 1659 1326 3198 1376

ny 3p, (2IB2) C2" scaled 2456 2760 1831 2093 1118 1237 1035 1176 2230 2508 494 449

exptlb 2782 1746 1500 1167 2843 1249 K

-

K*

ratioc 0.890 0.875 0.904 0.880 0.889 0.908

CzU 3190 1645 1410 403i 3291 964

(2IAl)

C2" 3294 935 1494 2504 3459 1061

scaled 2932 818 1351 2204 3075 963

-

K*

3155 1632 1410 530 3243 976

n,-r* C2" 3299 893 1397 1025i 3565 1023

(1 IA2)

ny-3s(l1B2)

scaled 2808 1428 1275 466 2883 886

C,

exptld 2847 1173 12909 683h 2968 898'

C21; 2324, 212U 841 1019 2195 763

C,

exptle 2275 1577 775

nu radical cation (2B2)

(llBI)

3242 878 1300 1022 3418 1024

scaled 2068 1862 760 897 1951 693

scaled 2885 768 1175 899 3039 930

C2" 3152 1726 1461 1261 3301 1075

scaled 2805 1510 1321 1110 2935 976

exptP 2560 1590 1210

Frequencies in inverse centimeters. The ratios between the experimental and R H F frequencies for the ground state are used as scaling factors to obtain the scaled frequencies for each excited state (see text). RHF, CIS, and UHF methods were used for the ground, excited, and radical cation states, respectively. * Reference 28. Experimental frequency/calculated frequency. Reference 27a as cited in ref 15. Reference 15. f From a normal coordinate analysis, these frequencies are an,equal mix of the V I and v2 ground-state vibrations. Reference 27c. Reference 27b, defined as the energy between the average of v(O+,O-) v(lt,l-). Reference 27d. j Reference 22.

-

with the experimental values, except for the C-0 stretch ( ~ 2 ) . The underestimation of the C-0 bond length24(see above) will cause the force constant and the corresponding frequency to be too large. This effect is observed. The out-of-plane bending frequency (v4) is a bit difficult to assign because of the inversion doublet splitting observed in the experimental spectrum. Since the nu ?y* structure is not rigidly bent, and then Innes and ~o-workers2~~ have shown that v4 will be split from being degenerate in the C, structure. As a consequence, using Brand's notation,17Cvibrationsof v4 will bedesignated, in increasing energy, by 0+, 0-, 1+, 1-, 2+, 2-, and so on. Our calculated vibrational frequency would correspond to the coalescence of the O+ and 0bands, so in Table V we have averaged the published frequencies of the (O+,O-) and (1t,l-) bands in order to obtain an experimental estimate which would be appropriate for comparison. From Jones and Coon's data,27bwe obtain a v4 of 683 cm-I, which is higher than our calculated value of 466 cm-l. It is interesting to note that this band is quite anharmonic as the experimental values for D2CO are 494 and 680 cm-1 for the average (O+,O-) (l+,l-) and (l+,l-) (2+,2-) frequencies, respectively. Experimental vibrational frequencies are also available for the n, 3s (1B2) state15 and the n, (2B2) radical cation.22 Our calculated frequencies agreevery well with the experimental values fortheZB2radicalcation,evenfortheC-Ostretch(~2).Therefore, the calculated C-O bond length for the 2B2 state is likely to be quite reasonable. The vibrational frequencies for the n, 3s state are in good agreement, especially for v3. We should note that the displacement vectors for the A I vibrations of the n, 3s state do show some mode mixing. From a normal coordinate analysis (see below for details), the n, 3s excited-state frequencies V I and v2 in H2CO are equal mixtures of the C-H and C-0 stretches in the ground state. The calculated (scaled) frequencies for V I and v 2 of H2CO are 2068 and 1862 cm-I, while frequencies have been observed at 2275 and 1577 cm-l.I5 There is no mode mixing for V I and v2 in DzCO; the calculated (scaled) values for the C-H and C-O stretches are 1378 and 1990 cm-1, respectively, and the experimental values are 1283 and 1701 cm-I.l5 Thus, the relative trends between isotopmers are well reproduced at the CIS level. The ny 3s state was calculated to have a shorter C-O bond length than the ground state, while the 2B2radical cation had essentially no change in this parameter. The C-H stretching frequencies for the Rydberg states decrease significantly as compared to the ground state or to the other excited states. This is expected as the C-H bond lengths are lengthened by 0.06 A from the ground state. We should point out that the trends between the experimental vibrational fre-

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-

-

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quencies of the n, radical cation and the nv 3s Rydberg state are also evident in the calculated frequencies. We have also listed the vibrational frequencies for the ny 3p,, A A * , and n, A* states. While there are no experimental data with which to compare, it is worth noting that the A A* and n, A* states have vibrational frequencies very different from those of the ground state or any of the other excited states. In particular, the lengthened C-O bond length (- 1.45A) causes an upheaval in the energetic ordering of the vibrational modes, and for instance, the A A* C-0 stretching frequency is the lowest calculated frequency (8 18cm-I). Then, A* state shows similar behavior.

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Vertical Triplet Transition Energies There has been extensive experimental study of the triplet states of formaldehyde, and many vertical transition energies for these states have been assigned.30 We have also calculated the vertical triplet transition energies for formaldehyde with the 6-31 1(2+,2+)G** basis set and the singlet ground-state geometry. The CIS-MP2 energies are listed in Table 111. We have assigned the vertical triplet states via their charge density difference plots with respect to the ground-state distribution, and they are shown in Figure 4. As is experimentally observed, the two lowest triplet states correspond ton, A* and A A* transition^.^ At the CIS level the transition energies for these states are within 0.8 eV of the experimental values. The CIS-MP2 correction increases the transition energies, but they remain within 0.8 eV of the experimental values. Figure 4 also shows a variety of triplet Rydberg states for formaldehyde, and they were easily assigned. As with the singlet states, CIS-MP2 correction for the triplet Rydberg states provides excellent agreement (0.2-0.4 eV) with the experimental v a l ~ e s . ~The J~ energetic ordering of the states is duplicated at both the CIS and CIS-MP2 levels. In Table I11 we have listed the previous calculations6a-cJ,1,'6 of the vertical triplet transition energies of formaldehyde as well as the CIS-MP2/6-311(2+,2+)G** values and the experimental data.4J0 The CIS-MP2 method provides the best agreement for the triplet Rydberg states, and the transition energies are less than 0.2 eV away from the experimental values. The CIS-MP2 vertical transition energies for the triplet valence states are not as good as those from the previous methods, and the CIS-MP2 values differ by -0.8 eV from the experimental energies. The CIS-MP2 method, though, shows excellent agreement with the experimental values for the triplet Rydberg states.

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2%

0:

3 p I (7)

BaB: o r

33A2 ~ c 1 2 - a ' ~ , - 3 d , ~( 8 1

-

3 d 1 Z m Y 2 (91

Figure 4. Charge density differenceplots for the first nine vertical triplet excited states of formaldehyde at the CIS/6-3 1 1(2+,2+)G** level with respect to the ground state. The outermost contour is 0.0001 e/B3.

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From Figure 4 one observes that the 13Al state (T T * ) is now distinct from FA1(ny-3py) as theenergyseparation between the valence state and the Rydberg state is now larger than in the singlet states. As a consequence the compactness of the valence state is more visible. In addition there is a compact state (33A2) which is a mixture of T C H ~ T* and ny 3d,, configurations that can mix due to symmetry. In order to obtain some measure of the charge reorganization, we have simply summed all of the positive contributions to the charge density differenceplot. It is interesting to note that the valence and Rydberg states differ in the magnitude of this difference sum. For instance, the valence triplet states (1,2,4, and 8) have difference sums of 0.79,0.59, 0.75, and 0.70 electron, respectively. The Rydberg states (3, 5 , 6,7,9, and lo), on the other hand, have 0.95,0.97, 1.05, 1.05, 1.OO, and 1.09 electrons, respectively, emphasizing the single excited electron in the atomic-orbital-like Rydberg states.

-

-

Adiabatic Triplet Excited States

-

As for the singlet states there is some experimental information on the geometries of the lowest triplet valence states, 13A2(ny T * ) ~ I and 13A1(T- 7r*).30 Experimentally, the 3(ny- T * ) state is known to be bent in the excited state by -38O (T(H-C-O-H) = 142O).3' Using the CIS/6-311+G* level, we have examined the adiabatic geometries for both states. The geometrical information is listed in Table VI. Let us first consider the 13A23(ny T * ) state. At the CIS level optimization under Cb symmetry yielded a structure with a slightly elongated C-O bond (1.25 A). This structure was a transition state, and the vibrational mode corresponding to the imaginary frequencywas the outsf-planevibration. Optimization under C,symmetry gave a bent structure with a puckering angle of 5 1O (T(H-C-O-H) = 129') and bond angles that are decreased from 120'. Unlike previouscalculations,I6theCIS method shows that the adiabatic 3(ny T * ) state is actually more bent than the singlet state, and this is in agreement with experiment.l5We have also investigated the 13Azstate at the UHF/6-311+G* level, and we found behavior similar to that for the CIS method. Comparisonto experiment did reveal some interestinginformation. The CIS method yields a short C-O bond length by -0.03 8, while UHF has the bond length too long by -0.05 A. The puckering angle is well reproduced by both methods, but CIS exaggerates the bending. Inclusion of electron correlation at the UMP2/6-3 11+G* level yields a C-O bond length in between the

-

-

CIS and UHFvalues, but still longer than the experimental value. In general most of the geometric parameters at the UMP2 level are intermediate between the values from the CIS and UHF methods. The adiabatic transition energy is 3.63,1.99, and 3.52 eV at the CIS, UHF, and UMP2 levels, respectively, while the experimental value is 3.1 3 eV.31 The calculated difference between the vertical and adiabatic transition energies at the CIS level is 0.18 eV, and the experimental value is 0.37 eV.31 We have also investigated the geometry of the 13A,3(x+ T * ) state at the CIS level. Optimization under Cbsymmetry afforded an elongated C-O bond length (1.40 A) and some changes in the bond angles. This CzUstructure is a transition state. Following the mode for the imaginary frequency yielded a Csstructure with a 42O puckering angle ( 7 = 138O). The H-C-H angle is similar to that in the ground state, but the H-C-O angle is significantly decreased to 1 13O. Taylor, Wilden, and Comer30have estimated the C-0 bond length by assuming that the H-C-H angle is unchanged in the excited state as compared to the ground state so that the stretching potential of NO could be used. With their assumptions, they obtain a value of 1.423 A for the C-O bond length, in good agreement with our value of 1.405 A. We do note that our calculated H-C-H angle is only 2' larger in the l3AI 3 ( ~ - T * ) state than in theground state. The adiabatic transition energy is calculated to be 3.64 eV (4.83 eV, e~petimental).~~ The calculateddifference between thevertical and adiabatic transition energies for this state at the CIS level is 1.1 1 eV, which is in good agreement with the experimental value (1.03 eV).30 In Table VII, we have listed the calculated vibrational frequencies for the 13A2(ny.+ T * ) and 13AI(T T * ) states. Using the ground-state scaling factors at the RHF/6-311+G* level, we have scaled the calculated CIS and UHF frequencies for the triplet excited states after careful assignment of the vibrational modes with respect to those of the ground state. We find generally good agreement with the experimentalvibrational f r e q u e n c i e ~ . ~As ~ , would ~ ~ be expected for the 13A2(ny T * ) state, the CIS frequency for the C-O stretch is too high and the UHF frequency is too low as the bond lengths differ from experimental values in opposite directions. The 13Al(T T * ) state has only one experimental frequency, and it is the C-O stretch (Q). Our calculated C-O bond length is slightly short compared to Taylor, Wilden, and Comer's estimate, and the calculated frequency (1 164 cm-l) is therefore too large as compared to experiment (887 c ~ - I ) . ~ O

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Atomic Populations and Covalent Bond Orders One of our major interests is the degree of charge reorganization that occurs upon electronic excitation. It would be very useful to have some quantitative data on the amount of charge that is transferred in an ny T* transition and also whether a T T* transition is different. In addition, the Rydberg states seem to bear a close similarity to the corresponding radical cation, but are the atomic populations and other properties derived from the respective wave functions also similar? We have, therefore, investigated the atomic contributionsto the molecular properties according to Bader's theory of atoms in molecules.33 This method, using only the electronic wave function, allows for a unique partitioning of molecular properties into atomic contributions. Also, Cioslowski and M i ~ o have n ~ ~recently developed a method for obtaininga measure of the covalent bond order between atoms as defined in Bader's theory. Thus, we will be able to obtain a measure of the charge transfer that occurs upon electronic excitation and the degree of covalency between atoms. In Table VI11 we havelisted theatomic populations andcovalent bond orders for a variety of vertical states of formaldehyde with the 6-311(2+,2+)G** basis set and using the MP2/6-31G* geometry of the ground state. First of all, let us consider the change from the ground state to the 2B2 (ny) and 2BI( T ) radical cations. The ground state has a strongly polarized C-O bond

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Hadad et al.

4300 The Journal of Physical Chemistry, Vol. 97,No. 17, 1993

-

TABLE VI: Geometrical Parameters of the Triplet States of Formaldehyde (6-311+G*)s 3(ny

CIS parameter

3A2(C2J

3A" (C,)

3A2(C2J

1.2512 1.0791 122.47 118.77 180. 1 705.41' 0.762 38 3.75

1.2479 1.0959 111.47 113.84 129.26 0 818.5 0.766 64 3.63

1.3328 1.0700 126.5 116.75 180. 1 750.21' 0.823 66 2.08

energy' transn energyg

r*)

UHF

4C-O) r(C-H) L(H-C-H) L(H-C-0) r(H-C-0-H) NIMAG V

-

3A'' (C,) 1.3314 1.0788 118.4 113.2 138.2 0 894.4 0.827 02, 1.99

'(7

UMP2 'A' (CJ 1.3174 1.0915 117.02 112.96 135.68 d d 1.142 99 3.52

r*)

CIS 'A" (exptlb) 1.28 1.10 116

3AI (C2J 1.3916 1.0670 126.30 116.85 180. 1 692.31 0.763 44 3.72

142

3.13

'A' (C,) 1.4050 1.0733 118.72 112.89 138.1 1 0 869.3 0.766 43 3.64

exptlC 1.423

4.83

All bond distances in angstroms and angles in degrees. Reference 31. Reference 30. Vibrational frequencies at this level were not calculated. Absolute energy is -(I 13 value), in hartrees. 1(S2): 2.0154. In electronvolts.

+

-

TABLE VII: Vibrational Frequenciess of Formaldehyde Triplet Excited States 3(ny

r*) 3A2

CIS AI

VI

~2 ~3

BI B2

v4 ~5 V6

3(7r

UHF

-

T*)

SAl

CIS

c2ti

CS

scaled

C2"

CS

scaled

exptlb

C2"

cs

scaled

3239 1654 1407 705i 3350 946

3034 1606 1384 819 3110 857

2700 1405 1251 721 2765 778

3333 1261 1534 750i 3485 1046

3242 1264 1536 894 3365 1071

2885 1106 1389 787 2991 972

287lC 1218d

3353 1417 1599 6931' 3514 1253

3301 1330 1620 869 3424 1276

2938 1164 1464 765 3044 1159

524

exptlb 887

Frequencies in inverse centimeters. The ratios between the experimental and RHF frequencies for the ground state are used as scaling factors to obtain the scaled frequencies for each excited state (see text). Reference 30. Reference 32a. Reference 32b determined a value of 1281 cm-1.

TABLE VIE Formaldehyde Atomic Populations and Covalent Bond Ordersa atomic populationb state 1Al GS 2B2 ny RC 2 B T~ RC 1A2 ny 7r* (1) 1B2 ny 3s (2) 2B2 ny 3p, (3) 2Al T T* (4) 'Al ny 3py (5) l B I n, 7r* (6)

----

-

a

0.43 1 0.269 0.105 1.025 0.263 0.279 0.521 0.381 1.069

covalent bond order

0

C total 4.833 5.037 4.920 5.438 5.102 5.144 5.170 5.081 5.467

7r

1.541 1.724 0.892 1.890 1.732 1.716 1.425 1.573 1.844

H total 9.207 8.582 8.595 8.821 9.073 9.249 9.063 8.811 8.674

7r

0.014 0.004 0.001 0.043 0.003 0.003 0.028 0.023 0.043

C-O total 0.980 0.690 0.742 0.871 0.910 0.803 0.883 1.053 0.930

7r

0.664 0.444 0.094 0.274 0.495 0.513 0.171 0.262 0.263

C-H total 1.430 1.177 0.871 1.022 1.350 1.418 0.957 1.059 0.797

7r

0.006 0.001 0.000 0.040 0.001 0.001 0.01 1 0.005 0.041

total 0.958 0.824 0.886 0.867 0.753 0.760 0.867 0.808 0.943

Using the MP2/6-31G* geometry and the 6-31 1(2+,2+)G** basis set. The wave functions are R H F for the GS (ground state), UHF for the RC (radical cations), and CIS for the singlet excited states (ES). Integrated electron sums for each state are the following: GS, 16.000; 2B2RC, 15.000; 2 B RC, ~ 14.999; ESl, 16.001; ES2, 15.996; ES 3, 15.999; ES4, 16.000; ES5, 15.997; and ES6, 16.001 electrons. In units of electrons.

(C+-0-), and the ramifications of this have been discussed with regard to the stability of carbonyl groups.* The hydrogens in the ground state of formaldehyde are charge neutral. Then, radical cation shows that 0.62 e is taken from 0 (even though 0.18 e is gained in the ?F system), while 0.29 e is lost from H. The C atom actually gains charge (0.20 e) on going to the n, radical cation, despite a loss of a density. The ?F radical cation also shows a large loss of charge for 0 and for H (0.78 and 0.42 e, respectively). This effect is even more striking when one observes that the a population on 0 actually increases by 0.44 e on going to the a radical cation. There is a large gain for the C atom (0.62 e), and most of this increase is from the ST system. Thus, there is a large degree of charge transfer to C in both radical cations. These effects can be easily visualized by charge density difference plots of each radical cation with respect to the ground state. These plots of the u and a systems for the n, and ?F radical cations areshown in Figure 5. Solid (positive)contours correspond to charge accumulation in the radical cation as compared to the ground state, while dashed (negative) contours correspond to charge depletion. (While Figure 5 depicts the charge density difference plots using the RHF and UHF densities for the ground

and radical cation states, respectively, similar plots were also obtained from the corresponding MP2 densities.) The charge density difference plots clearly show that charge density is accumulated on the C atom in both radical cation states, but there are subtle differences. The n, radical cation shows an accumulation of charge on the carbon atom in the u system, and the lone pairs on the 0 atoms are depleted. The 0 atom gains a charge in the radical cation. In general, the H atoms lose charge. The difference plots for the a radical cation show a smaller depletion of the lone pairs on the 0 atom and general depletion for the ?r system. There is an overall loss of charge in a diffuse sense for both radical cations, and this would correspond to dispersal of the charge to minimize the electrostatic energy. However, near the carbon and oxygen, there is an accumulation of charge. The integrated atomic populations in Table VI11 reveal the same trends for the u, x , and total charge shifts. The covalent bond orders also show that there is a decreased bonding interaction between C-O and C-H on going to thevertical radical cations. The loss of C-0 covalent bond order is greater for the ny radical cation, while the a radical cation has a larger


The Journal of Physical Chemistry, Vol. 97,No. 17, 1993 4301 therefore, are very dependent on not just the termination state (for instance, a*)but also the originating molecular orbital. This is easily seen on comparing the n, a* and n, A* excited states. However, it is quite easy to distinguish the valence states from the Rydberg states because of the latter’s similarity to the corresponding radical cation.

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Acetaldehyde The ground state of acetaldehyde has been well studied, leading to the torsional barrier of the methyl as well as an analysis of the vibrational f r e q ~ e n c i e s .The ~ ~ n A* excited state has also been examined in order to probe the change in the torsional barrier upon electronic excitation37as well as the dissociation dynamicP and photolysis products.39 Acetaldehyde has been shown to possess a very long p Rydberg series up to n 45,40 in contrast to formaldehydeand acetonewhich show resolvable series to n = 12. It is known that, similar to formaldehyde, the n A* state is pyramidally distorted in the excited state, and the barrier to methyl rotation is increased significantly in the n a* excited state. The n 3s state also has a larger barrier than the ground state, but experimentally, it is known that the s-cis (Cs,eclipsed) conformation of the ground state is An examinationof the Rydberg and valence states of acetaldehyde will provide interesting information about the differences in these states that cause such different behavior.

-

281

8

RC

Id

syrlem)

281 r RC ( n r y r l e m )

Figure 5. Charge density difference plots for the vertical (a) ny and (b) T radical cations with respect to the ground state for the molecular plane ( u ) and the plane containing the C-O bond but perpendicular to the molecular plane ( T ) using the 6-311(2+,2+)G** basis set. The molecule is drawn to scale, and the oxygen atom is to the right. The outer contour is 0.0001 e/B3, and the contours increase by a factor of 2.

change in the C-H bond order. These changes would therefore explain the geometric distortions noted above for the adiabatic geometries. Let us first examine the Rydberg states. In Table VI11 one can see that the a populations for each atom in the n, 3s and n, 3p, states are very similar to that of the n, radical cation. The total populations are quite different, but they must be since there is one less electron in the radical cation. The A covalent bond orders are also similar for these states and for the correspondingradical cation. In comparison to the ground state, the ny 3s and n, 3p, states show only a small change in the 0 atomic charge (-0.1 3 and +0.04 e, respectively). Both Rydberg states suffer charge loss at the H atoms, and the C atom gains charge as compared to the ground state. The bond orders for both Rydberg states do not change significantly from the groundstate values, but the C-H bonds become weaker in the Rydberg states. This supports the observation noted above for the lower C-H stretching frequencies in the adiabatic Rydberg states. The 3Al (predominantly n, 3p,) state is not very similar to the nu radical cation because as mentioned above, the 3A1 state is a mixture of a a* and ny 3p, configurations. The a A* contribution to the overall state makes the corresponding properties dissimilar to those of the n, radical cation. Let us now consider the valence states. The ny a*, (predominantly) a a*, and n, A* states show behavior much different from that of any of the radical cations. The n, a* state shows that 0.39 and 0.11 e is lost from 0 and H, respectively, the C gains 0.61 e as compared to the ground state. The n, A* state is similar with 0.53 and 0.05 e lost by C and H, and 0.63 e gained by C. The changes in the a a* state are somewhat smaller in magnitude as 0 and H lose 0.14 and 0.34 e, respectively, and C gains 0.34 e. Also the n, a* and n, u* states have a large amount of a charge being concentrated on C, but there is also an additional a electron for these states as compared to the u a* state. The covalent bond orders show that there is a significant decrease in the C-O bond order for each of the three states. The a bond order has also decreased (-0.26 for both n a* states and 0.17 for the a a* state). It is interesting to note, though, that each of the valence states is quite distinct, and while there are some similarities(for example, similar a bond orders), there is not overall agreement among all of the valence states. The properties of the valence states,

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Vertical Singlet Excited States We have calculated many of the vertical singlet excited states of acetaldehyde using the MP2/6-3 lG* geometry for the ground state, which is similar to the experimental geometry (see bel0w).~2 The vertical excitation energies and our theoretical assignments arelistedinTableIXforthe6-311(2+)GS basisset. Our method of assignment of the states is via our charge density difference plots, and these plots are shown in Figure 6 for the first 18 vertical excited states. Once again, solid (positive) contours correspond to charge accumulation in the excited states as compared to the ground state, and dashed (negative) contours correspond to charge depletion. From Table IX we see that the CIS method provides good agreement for then A* valence state with the transition energy -0.6 eV away from the experimental ~ a l u e . This ~ , ~state ~ is not strongly affected by the more flexible basis set, but one diffuse function is needed. The transition energies for the Rydberg states are calculated to be 1.5 eV too high at the CIS level. The CIS-MP2 correction affects the transition energies for the Rydberg states significantly more than those for thevalence states, and at the CIS-MP2 level our theoretical transition energies are within -0.4 eV for all of the Rydberg states for which experimental transition energies are known. Agreement for the n a* state is 1 eV at the CIS-MP2 level. The experimental energetic ordering of the states, though, is well reproduced at the CIS and CIS-MP2 levels. From Figure 6 one can easily see the valence states of acetaldehyde (1,6, and 7). There are many excitations from the highest occupied MO (n) as well as excitationsfrom other orbitals. Plots of )I for the six highest occupied molecular orbitals of acetaldehyde, as well as our nomenclature for describing them, are shown in Figure 7.44 Since acetaldehyde is only of C, symmetry, there are many transitions that are of the same symmetry, and therefore some configurational mixing can occur. We saw this effect earlier for the 2IAl and 3IAl states of formaldehyde, and it is more pronounced in acetaldehyde. For instance, state 7 (3A”) is a mixture of excitations from occupied molecular orbitals 7, 8, and 10 to the a* orbital. Also, state 6 (5A’) is predominantly the A a* transition, but there is some admixture of the n 3px as well. Therefore, state 4 (4A’) is predominantly the n 3px configuration, but the minor component is the a a* configuration. Examination of the

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Hadad et al.


TABLE I X Vertical Transition Energies for Acetaldehyde. CIS/6-311(2+)G* cm-1 (x 10-3)

assignment

------ --

2 3 4 5 6 7 8 9 10

1A” n 2A’ n 3A’ n 4A’ n 2A“ n 5A’ T 3A“ 6 7 , 6A‘ n 7A‘ n 4A” n

1 2 3 4 5 6 7 8 9 10

1A” n T * 1A’ T T * 2A’ n 3s 2A“ ~ 7 ~ , g ~, 3A’ n 3p, 3py 4A‘ n 3p, 3A” n 4A“ 6 7 , ug, ~ 5A’ n 3d,? 5A” T 3s

1

T*

3s 3py 3p, 3p,

+

T*

U S , 610

T*

3d,2 3d, 3dy,

39.5 68.7 74.4 75.0 75.5 78.5 78.9 82.2 82.7 83.2

osc str

CIS-MP2,b eV

exptl: eV

0 0.0105 0.1156 0.0492 0.0079 0.2949 0 0.0124 0.0028 0.0035

5.27d 6.71 7.57 8.00 7.37 9.07d 10.34d 8.09 8.08 8.10

4.28 6.82 7.46 7.75

4.93 7.37 6.84 9.74 7.56 7.52 7.61 10.35 8.17 10.24

3.97 5.999 6.81

b. Tripletf 4.15 4.99 8.24 8.60 8.93 9.25 9.30 10.01 10.03 10.13

+

1 0 T*

1 0

eV a. Singlet 4.89 8.51 9.22 9.30 9.37 9.73 9.78 10.19 10.26 10.31

a*

8.43 8.69e

7.44 7.80 8.40

Using the ground-state MP2/6-31G* geometry. The energies of the ground state at the RHF/6-31+G*, RHF/6-311(2+)G*, and MP2(full)/ 6-311(2+)G* levels are -152.918 863, -152.953 485, and -153.487 273 hartrees, respectively. Using the 6-311(2+)G* basis set and correlating all electrons. Reference 4, p 256. States 1 and 7 are valence states, and the CIS-MP2 correction increases the transition energies relative to CIS. State 6 is a mixed valence state, as some Rydberg character is present, and the transition energy decreases at CIS-MP2 compared to CIS (see text). Reference T * and K K* values are vertical; all others are adiabatic. Data are taken from ref 43. g Vertical energies for the T T* state are 43. f The n also reported at 6.31 and 6.25 eV (Van Veen, E. H.; Van Dijk, W. L.; Brongersma, H. H. Chem. Phys. 1976, 16, 337. Staley, R. H.; Harding, L. B.; Goddard, W. A.; Beauchamp, J. L. Chem. Phys. Lett. 1975, 36, 589).

-

-

-

charge density depletion regions (the negative contours) for the density difference plots is useful. These plots of the depletion regions for the vertical singlet excited states of acetaldehyde as well as for the vertical n radical cation are shown in Figure 8. States 1 (1A”, n r * )and 7 (3A”, u7,US, uIo T * ) are clearly different from the other states. States 4 (4A’) and 6 (5A’) are similar to each other but different from the radical cation. State 6 (5A’) shows a greater degree of perturbation near the nuclear centers and, therefore, has a greater degree of valence character than state 4. State 6 is also less sensitive to diffuse functions in the basis set than is state 4. If one examines the charge reorganization from the density difference plots (by summing the positive and negative contributions), one observes similar trends. States 1, 4, 6, and 7 have positive sums of 0.77,0.89, 0.61, and 0.89 electron, respectively,while states 2,3,5,8,9, and lOhave 1.04,0.99,1.15,1.12,1.18,and1.21 electrons,respectively. This sum clearly aids in differentiating the valence states from the Rydberg states. We have also examined the vertical IP of acetaldehyde using the 6-311(2+)G* basis set. The results are listed in Table I1 using the MP2/6-3 lG* geometry of the ground state. Weobtian good agreement with the experimental n vertical IP22.43with this basis set, especially when electron correlation is added (even only at the MP2 level). Finally, we point out that different states rearrange the charge density in different ways, and in particular, the n r* depletes a large amount of charge from the C-0 bond, while other states concentrate charge density in that region. The charge density around the methyl group is also very different in some states. This will have important ramifications for the adiabatic geometries.

-

-

-

Adiabatic Singlet Excited States We have examined the adiabatic geometries of some of the singlet excited states of acetaldehyde, and the geometrical parameters are listed in Table X. Some of the states considered here have been studied theoretically,45 although we are unaware of any study which performed complete geometry optimizations.

For our calculations, the 6-31+G* basis set was used for all of the optimizations, and a comparison of the RHF/6-31+G* geometry and the experimental geometry for the ground state4* is listed in Table X. One can see that the RHF level is quite good at predicting all of the geometric information for the ground state, except for the C-0 bond length which is calculated to be shorter than the experimental value. This is a known defect of Hartree-Fock theory for very polar bo11ds.2~ The charge density difference plots for the vertical n n* state would predict that the C-O bond length would increase in the excited state as compared to the ground state. Also, it is known that this state is pyramidalized about the C-0 bond in the excited ~ t a t e . ) ~We , ) ~have optimized the s-cis, C, (eclipsed) geometry of the n r* state at the CIS/6-31+G* level, and indeed, the C-0 bond length does lengthen by -0.08 A. The C-C bond length also increases but only by -0.02 A. There is a large increase in the C-C-Ha angle. Calculation of the vibrational frequencies of the C, structure shows that there are two imaginary vibrational frequencies. The vibrational mode corresponding to the larger of the two imaginary modes is the pyramidalization mode around the C-O bond. The smaller imaginary vibrational frequency corresponds to methyl rotation. The staggered (C,) orientation of the methyl group with respect to the C-O bond corresponds to a 0.05-eV decrease in energy, and this structure is a transition state at the CIS level. Removal of all symmetry constraints and geometry optimization along the larger imaginary vibrational mode from the eclipsed (C,) structure yielded a stationary point with zero imaginary vibrational frequencies. This structure has a pyramidalized carbonyl carbon with a dihedral angle 7(H,-C-O-C) of 145O. Noble and Lee used an assumed C-0 bond length of 1.316 A, a C-C-H angle of 113.5O, and a staggered H-C-C-0 arrangement (60O) to predict a 145’ pyramidalization angle.38 Our calculations show that the (Me)H-C-C-O dihedral angles are not quite staggered (60O) but differ slightly (-56O), while the C-C-H(Me) angles change only slightly from the ground state’s values. However, the angles around the carbonyl carbon are quite different from those in the ground state. In particular, the 0-C-C angle is

-

-



n

-

I'

11' ( I )

n ' 3 p y 31'

(3)

07

1

a

- 34,2

-

I'

(PI. 4d,?

[I)

0 8 8a' ( 8 )

IT^ l a ' ( 9 )

9.1'

[IO)

I

2s' (It)

o l o a ' (12)

51' ( 6 )

010

61' ( 8 )

Figure7. $for thesix highest occupied molecular orbitals of acetaldehyde at the 6-31 1(2+)G* level (0.05 au for the outer contour).

P

I *

74'

I'

11' ( I )

+

3s 21' (2)

II *

3 p j 31' (3)

n

1'

91' ( 1 4 )

O

Figure 6. Charge density difference plots for the first 18 vertical singlet excited states of acetaldehydeat the CIS/6-3 1 1(2+)G* level with respect to the ground state. The outermost contour is 0.0001 e/B3.

311 41' ( I )

-

' 1 7 , 0 ~ , 0 ~ fi'~

31' (7)

I

I;

+

3pz 21' (5)

3422 61' (8)

I

+

51' ( 6 )

21' Radical C i l i o I ;

118O in the CI structure and 124O in the ground state. Our Figure 8. Charge density depletion regions (negative contours) for the adiabatic transition energy is calculated to be 4.54 and 4.79 eV first eight vertical singlet excited states and the n radical cation of acetaldehyde (O.OOO1 e/B3). at the CIS and CIS-MP2//CIS levels, respectively (3.69 eV, experimental).38 Experimentally, the n 3s excited state is known to remain using the optimized geometries (and energies) for the eclipsed s-cis (eclipsed), similar to the ground state.41We have optimized and staggered conformations of the ground and n 3s Rydberg the n 3s excited state under Cssymmetry, and this structure excited states using the 6-3 1+G* basis set. For the ground state, is a minimum as verified by a vibrational frequency analysis. the calculated rotational barrier is 355 and 310 cm-I at the RHF There are some major distortions in the structure though. The and MP2 levels, respectively, and both values are in reasonable C-O and C-C bond lengths decrease and increase by 0.04 and agreement with experiment. Ozkabak and Goodman have 0.09 A, respectively, and there is a large increase in the C-Ha reported that the ground-state barrier is 373 and 364 cm-1at the bond length to 1.129 A. There are even larger changes in the RHF and MP2/6-3 1G** levels, respecti~ely,3~b and therefore, bond angles. The 0-C-C angle increases to 131O , and the C-Cmore balanced basis sets than 6-31+G* will yield even better H(Me) angles decrease to 104O. This effect would have large agreement with experiment. For the n 3s Rydberg state, the effects on the vibrational frequencies (see below). Our calculated calculated rotational barriers are 356 and 1065 cm-l at the CIS/ CIS adiabatic transition energy is 8.49 eV which is much larger 6-3 1+G* and CIS-MP2/6-31+G*//CIS/6-3 1+G* levels, rethan the experimental value (6.82 eV).46 As was seen above, a spectively, and the latter compares favorably to the experimental single point energy calculation at the CIS-MP2/6-31+G* level value (880cm-1).41b As was noted above, theCIS-MP2 correction with the CIS/6-3 1+G* geometry yields an adiabatic transition is quite important in order to obtain good agreement with the energy of 7.53 eV which is in better agreement with experiment. experimental energies for the Rydberg states, and the CIS-MP2 Experimentally, the methyl torsional barrier in the n 3s level does reproduce thequalitative trend that the barrier isgreater Rydberg state of acetaldehyde (880 ~ m - l )is~known ' ~ to be larger than that of the ground state (408 or 397 ~ m - ~ ) We . ~ have ~ ~ - ~ in the n 3s Rydberg state as compared to the ground state. We will discuss the origin of this larger barrier below. also calculated the effective rotational barriers of these states

-

-

-

-

-

-

-

4304

TABLE X

Geometries of the Ground and Excited States of Acetaldehyde (6-31+G*)a

ground state (1A') RHF MP2b exptlC parameter r(C-0) r(C-C) r(C-Ha) r(C-Hb) 4C-W r(C-Hd) L( 0-c-C) L(C-C-H,) L(C-C-Hb) L(C-C-H,) L(C-C-Hd) T ( Ha-C-0-C) T ( Hb-C-C-0) T( H,-C-C-0) T (Hd-C-C-0) NIMAG Y

c,

c,

1.1893 1.5031 1.0947 1.0817 1.0868 1.0868 124.48 115.62 110.62 109.64 109.64 180 0 121.28 121.28 0 150.5

1.2222 1.5017 1.1090 1.0900 1.0948 1.0948 124.39 115.27 109.84 109.86 109.86 180 0 121.17 121.17

CIS transn energyd CIS-MP2 transn energyd/ a

Hadad et al.


C, 1.213 1.504 1.106 1.091 1.085 1.085 124.0 114.9 110.6 110.3 110.3 180 0

n

C,(rcl) 1.2600 1.5250 1.0833 1.0816 1.0852 1.0852 121.29 121.80 109.48 110.77 110.77 180 0 1 19.69 119.69 2 369.01' 129.21' 4.87 4.93

(1A") CIS C, (stag) 1.2576 1.5227 1.0824 1.0821 1.0857 1.0857 120.12 122.26 108.06 111.22 111.22 180 180 60.48 -60.48 1 419.61'

-+

2A' (RC (ecl)

?r*

4.82 4.84

CIS

UHF

UMPZ

CI

C,(ecl)

C,

CI

c,

c,

1.2562 1.5309 1.0885 1.0824 1.0877 1.0821 118.34 117.03 110.00 111.59 108.11 145.39 -55.51 65.35 -174.53 0 202.0

1.1439 1.5959 1.1287 1.075 1 1.0839 1.0839 131.73 107.85 107.49 103.63 103.63 180 0 121.26 121.26 0 157.6

1.4585 1.4677 1.0740 1.0824 1.0944 1.0944 118.63 125.36 110.37 111.35 111.35 180 0 121.20 121.20 1 255.91'

1.4856 1.4719 1.0794 1.0814 1.0958 1.0896 112.80 121.82 112.12 105.35 113.21 136.50 -55.67 68.86 -174.36 0 234.3

1.2209 1.4892 1.0879 1.0795 1.0870 1.0870 122.09 123.60 111.96 107.19 107.19 180 0 122.22 122.22 0 138.0

1.2165 1.4929 1.1092 1.0885 1.0972 1.0972 123.17 122.35 1 1 1.83 106.61 106.61 180 0 122.46 122.46

4.54 4.79

8.49 7.53

8.61 8.34

8.04 8.30

8.86

10.411

e e

Bond distances in angstroms and angles in degrees. Absolute energies are the following: G S 1A' (RHF), -152.921 12 (Cs);1A" (CIS), -1 52.742 10

(C,(ecl)), -152.743 85 (C, (stag)), -152.754 23 (CI); 2A' (CIS), -152.608 98 (C,); 3A' (CIS), -152.604 78 (CJ, -152.625 55 (CI; 2A' RC (UHF), -152.595 52 (C,);and 2A' RC (UMP2), -152.987 84 (C,) hartrees. Using the 6-31G* basis set. Reference 42. In electronvolts. e Vibrational frequencies were not calculated. /The MPZ(fu11)/6-31+G*//RHF/6-31+G* and MP2(full)/6-3l+G*//MP2(full)/6-3 1+G* energies for the ground state are -153.368 06 and -153.370 48 hartrees, respectively.

We have also optimized the ZA' (n) radical cation using the UHF and UMPZ methods with the 6-3 1+G* basis set. The s-cis (Cs,eclipsed) n radical cation is a minimum at the UHF level, and the geometry is generally quite similar to that of the ground state, except for the C-C-H, angle which increases by -8'. The adiabatic IP is calculated to be 8.86 and 10.41 eV at the UHF and UMP2 (optimizedgeometry) levels. We havealsocalculated the adiabatic geometry of the staggered orientation of the methyl group in the n radical cation at the UHF and UMP2/6-31+G* levels. The effective rotational barrier is 388 and 408 cm-I, respectively, and the eclipsed conformation is more stable. These values are in reasonable agreement with previous estimates of 223 and 187 cm-1 at lower levels of t h e ~ r y . ~ ~ ~ . ~ The adiabatic geometry of the A A* state is also of some interest. Using the CIS/6-3 1+G* level, we have studied the 3A' state. Since the A A* configuration and a few of the n 3p Rydberg configurations are of the same symmetry, the character of the 3A' state will be mixed as discussed above. The mixing contribution will also be sensitive to geometry. For example, C, optimizationof the (predominantly) n- 3pystatefrom theMP2/ 6-31GS geometry of the ground state allowed a lengthening of the C-0 bond length to 1.459 A, and in particular the final state has changed to being predominantly a A A* transition. The occupied molecular orbitals are quite sensitive to geometry, and the A and n orbitals reverse their energetic ordering upon lengthening of the C-O bond. While the occupied n molecular orbital is higher than A in the ground state, the lengthened geometry has the ?r orbital higher in energy. The resulting A A* state has a very long C-0 bond length with large changes in the angles around the carbonyl carbon as compared to the ground state. For instance, the 0-C-C angle has decreased to 119' from 124' in the ground state, while the C-C-Ha angle has increased to 125' (ground state 116'). This C, geometry is a transition state with the imaginary vibrationalmode corresponding to the pyramidalization of the carbonyl carbon. Optimization under CI symmetry afforded a bent structure ( T = 137') with a rotated methyl group. The torsion angles of the methyl

-

-

-

-

-

-

hydrogens(r(H(Me)-C-C-O)

=-56', 69',and-l74O) arevery similar to the n A* values in its CIstructure (-56', 65O, and -175'). The calculated adiabatic transition energy for the ?r A* state is 8.04 eV at the CIS/6-31+G* level and 8.30 eV at the single point CIS-MP2 level. We have also examined the vibrational frequencies of the different states of acetaldehyde. The calculated ground-state vibrational frequencies of the do, dl, d3, and 4 isotopomers of acetaldehyde are given in the supplementary material along with the experimental assignment^.)^ Theratios between thecalculated frequencies at the RHF/6-3 1+G* level and the experimental values are also listed. The calculated vibrational frequencies for the deuterated isotopomerswere assigned via a normal coordinate analysis, followed by an analysis of the mode mixing.47 For example the calculated force constants were converted from Cartesian coordinates to symmetry coordinates (via the appropriate B matrix), and the B matrix was then used to obtain the G matrix according to Wilson's GF matrix f o r m a l i ~ m . A ~~ calculation of the vibrational frequencies produced the corresponding L matrix, which describes the form of the normal coordinates for the molecular vibrations. The relative mode mixing of one isotopomer (or molecular geometry) to the groundstate vibrational modes is determined according to the following rela tion:@ QD

-

= L-'&Q,

where QDand QH are the normal coordinates of the deuterated and nondeuterated isotopomers,respectively. The matrix product I!-]&, therefore, describes the degree of mode mixing from one isotopomer to another, or equally well, an excited state'svibrations in terms of the ground-state vibrational modes. (These matrix products are provided as supplementary material.) We shall use this method of assignment below for comparison of the excited states to the ground state, but we note that there is significant mode mixing of the low-energy vibrations (VS-vs and V I ~ - V , ~ in ) the deuterated isotopomers of the ground state as well as in the excited states.



TABLE XI: Average Scale Factors for Acetaldehyde Ground-State Vibrational Frequencies vibration three methyl C-H(D) stretches ( Y I , one aldehyde C-H(D) stretch ( 4 one C-0 stretch ( 4 eight bending modes ( Y S - U S , Y12-tq4) two torsional modes ( Y I O , Y I ) )

CH3CHO

CH3CDO

CD3CHO

CD3CDO

0.91 1 0.861 0.868 0.893 0.943

0.91 1 0.881 0.875 0.898 0.949

0.921 0.861 0.874 0.896 0.949

0.922 0.879 0.874 0.899 0.970

v2, Y I I )

-

TABLE XIk Vibrational Freauencies’ of Acetaldehyde Valence Singlet Excited States (CIS/6-31+GS) ~~~

n mode A’

VI Y2

v3 v4

v5 Y6

Yl

vs v9 VI0

A“

VI I

VI2

VI3 VI4

VI5

A*

(1IA”)

cs

CI

scaled

3308 3198 3263 1689 1619 1291 1523 1082 977 42 1 3251 1610 1105 3691 129i

3313 3187 3202 1672 1607 1277 1508 1092 968 408 327 1 1617 1123 556 202

3018 2903 2757 1451 1435 1140 1347 975 864 385 2980 1444 1003 497 190

(3IA’)

A-A*

exptlb

1120

370

454c 195

CS

CI

scaled

3292 3109 3395 1119 1585 1384 1517 1216 849 432 3237 1566 2768 767 2561’

3302 3120 3330 1265 1543 1046 1506 1210 811 404 3190 1615 763 1093 234

3008 2842 2867 1098 1378 934 1345 1081 724 381 2906 1442 68 1 976 221

Frequencies in inverse centimeters. The average scaling factors between the experimental and RHF frequencies for the ground state are used to obtain the scaled frequencies for each state (see text). Reference 51. Determined as the differencebetween the 140’and 1402experimental assignments.

As for the ground state, the calculated frequencies of the excited states will have to be scaled by -0.9 in order to compare with e ~ p e r i m e n t .Due ~ ~ to the large mode mixing, we have determined average scale factors for the ground-state isotopomers, and they are listed in Table XI. We have chosen to separate the vibrations into five groups: methyl C-H(D) stretches, aldehyde C-H(D) stretch, C-0 stretch, bending modes, and low-energy torsional modes. One should note that the bond stretches around the carbonyl carbon have the smallest scale factor, as the calculated bond lengths are too short and the corresponding force constants and vibrational frequencies are too large. The scale factors also increase with deuterium substitution due to the decreased anharmonicity in the C-D experimentalvibrational freq~encies.~~ Let us first consider the valence excited states of acetaldehyde. The n A* state has been extensively studied, and some experimental frequencies are known.51 In Table XI1 we have listed the calculated frequencies at the CIS level for both the C, (eclipsed) and CI geometries of the n A* state. Using the average scaling factors discussed above (Table XII), we have also listed the scaled frequencies for the Cl structure. We find good agreement for the low-energy torsional modes of the n A* state, but the C-0 stretch (v4) is overestimated. This is expected since as in formaldehyde, the CIS optimized geometry tends to predict too short a C-0 bond length, and therefore, the stretching frequency will be exaggerated. Unfortunately there are no experimental data for the A A* state, and in general the frequencies are similar to those of the n A* state, except that the C-0 stretching frequency ( ~ 4 is ) much lower for the A A* state. There are much more data available for the vibrational frequencies of then 3s state of a ~ e t a l d e h y d e .In~ particular ~~~~ the different isotopomers (do,dl, &, and d4) have all been studied, and this will allow an excellent comparison to our calculated values. Since the optimized geometry for the n 3s state is significantly different from that of the ground state, there is a large degree of mixing in the bending modes of acetaldehyde. In addition, each isotopomer shows different mixing, and the relative mode mixing (in comparison to the ground-state normal coordinates) is given in the supplementary material. In Table XI11 we have listed the frequencies for each isotopomer and attempted to assign the calculated values to the modes of the ground state.

-

-

-

-

- -

- -

-

However, the extensivemixing of Q - Y ~ and YI makes a unique correspondence to the ground-state frequencies very difficult. We note that the normal coordinate assignment was essential in providing at least a consistent ordering. The average scaling factors in Table XI were also applied, and the scaled frequencies are listed in Table XIII, along with the available experimental data for the n 3s state. Many of the experimental frequencies were generally ordered by numerical value and then assigned to a particular vibrational mode by comparison to the ground-state frequencies.46.52 We have listed the experimental values in Table XI11 according to numerical agreement with our calculated (scaled) frequencies. In general we obtain very good agreement between our calculated (scaled) frequencies and the experimental ones.46-52 Our agreement is within 60 cm-I for most cases, except for the C-H(D) stretch (v3) where the calculated frequency is too low. The calculated aldehydic C-H bond length must therefore be too long by a small amount. However, we emphasize that we obtain very good agreement for most of the vibrational frequencies. The relative trends between isotopomersis also well reproduced by the calculated frequencies. In addition, the large deuterium isotope effect for the aldehydic C-H stretch (vj) in the n 3s state46953 is in accord with our calculated vibrational frequencies and also the large change in the aldehydic C-H bond length in the adiabatic state. Finally, we note that our calculated frequencies are significantly different from those calculated by Crighton and Bell using their SCF geometry.45 We have similarly listed the vibrational frequencies for the 2A’ (n) radical cation obtained at the UHF/6-31G* level as well as the experimental values54 in Table XIV. As with the n 3s state, we find good agreement for all of the bending and torsional vibrations, but not for the aldehydic C-H(D) stretch. However, in this case the C-H(D) stretching frequency for the radical cation is too high, showing that the UHF level predicts too short a C-H bond length for the radical cation. We obtain good agreement for all of the isotopomers.

-

-

Vertical Triplet Transition Energies Using the MP2/6-3 lG* geometry of the ground state, we have calculated the vertical transition energies for the triplet states at the CIS level using a variety of basis sets. The excitation energies

4306 The Journal of Physical Chemistry, Vol. 97, No. 17, 1993 TABLE XIII: Vibrational Frequencies' of the Acetaldehyde 2'A' n mode

CIS

scaled

3398 3189 2609 2144 1529 998 1415 1064 530 32 1 3311 1552 1137 832 158

3096 2905 2246 1861 1365 89 1 1264 950 473 303 3016 1386 1015 743 149

exptlb

CIS

scaled

3398 3189 1873 2166 1528 776 1396 1030 519 317 3311 1552 1109 721 151

3096 2905 1650 1895 1372 697 1254 925 466 30 1 3016 1394 996 647 143

-

Hadad et al.

3s Rydberg State (cIs/631+G*)

exptlb

CIS

scaled

2517 2282 2615 2139 1097 947 1169 775 552 305 2456 1126 1029 670 128

2318 2102 2252 1869 983 849 1047 694 495 289 2262 1009 922 600 121

~~~~

A'

VI

y2

v3 v4 VS

v6 Vl

v8 v9

VI0

A"

VII

~ 1 2 ~ 1 3 VI4

VIS

2452 1291 7628 1156 90 1 321

657 227

exptlc

CIS

scaled

2524 2282 1866 2162 1119 732 1062 800 513 302 2456 1126 957 628 120

2327-2104 1640 1890 1006 658 955 719 46 1 293 2264 1012 860 565 116

exptlb

~~~

1888c 1324 792c8 118Y 826f 327

219

2472 983 869/ 1103c 6848 314

1864d 994 649 876f 807 300

171

Frequencies in inverse centimeters. The average scaling factors between the experimental and RHF frequencies for the ground state are used to obtain the scaled frequencies for each state (see text). Reference 52. Reference 46. Assigned as v3. Assigned as V6. /Assigned as v7. g Assigned as v9.

TABLE XIV: Vibrational Freauencies. of Acetaldehvde *A' Radical Cation (UHF/631+G*) mode A'

VI

v2 y3 v4 VS

v6

Vl

v8 v9 VI0

A''

VII

U I ~

VI^ VI4

VIS

.

CH3CHO

UHF

scaled

3358 3203 3265 1709 1576 1352 1485 1185 93 1 457 3277 1571 1217 838 138

3059 2918 2811 1483 1407 1207 1326 1058 83 1 43 1 2985 1403 1087 748 130

CH3CDO exptlb

2570

126oC llOod 770

UHF

scaled

3358 3203 2406 1692 1576 1071 1483 1189 850 452 3277 1568 1152 728 134

3059 2918 2120 1481 1415 962 1332 1068 763 429 2985 1408 1034 654 127

CD3CHO exptlb

2020

129oC 102od 770

UHF

scaled

2495 2299 3265 1683 1099 1344 1206 1023 81 1 420 2426 1105 1172 687 110

2298 2117 2811 1471 985 1204 1081 917 727 399 2234 990 1050 616 104

CD3CDO exptlb

2590

l05oC 750 330

UHF

scaled

2495 2298 2406 1666 1099 1210 1044 1000 77 1 44 1 2426 1133 1031 619 105

2300 2119 2115 1456 988 1088 939 899 693 428 2237 1019 927 556 102

exptlb

1890

96oC 83oC 320

Frequencies in inverse centimeters. The average scaling factors between the experimental and RHF frequencies for the ground state are used to obtain the scaled frequencies for each state (see text). Reference 46. Assigned as US. Assigned as V6. Assigned as v7.

are listed in Table IX, and the charge density difference plots for the first nine vertical triplet states at the CIS/6-3 11(2+)G* level are shown in Figure 9. Once again, the density difference plots are with respect to the ground-state charge distribution, and positive (solid) contours correspond to charge accumulation in the excited states as compared to the ground state. The 3-D contour plots make the assignment of the excited states quite trivial, and inspection of Figure 9 shows many valence states (1, 2,4, and 8) and s, p, and d atomic-orbital-like Rydberg states. The valence states are more compact than their Rydberg counterparts, and moreover, the triplet Rydberg states appear very similar to the singlet excited states. As a result, the valence states are less sensitive to flexibility in the basis set. Comparison to the experimental values of Kupperman and co-workers43shows that the CIS method provides reasonable values for the 3(n r * )and 3(7r r * )valence states, but the calculated values for the Rydberg states are high by 1.4 eV. This was also observed above for the singlet states. Further inclusion of electron correlation at the CIS-MP2/6-3 11(2+)G* level improved the agreement with experiment for the Rydberg states to within -0.3 eV. The transition energies for the valence states increase at the CIS-MP2 level as compared to the CIS values. The relative charge reorganization is again quite diagnostic of valence vs Rydberg transitions. Using the simple summation of the positive charge shifts for each state with respect to the ground state yields a quantitative difference between the two classes of states. In particular, states 1, 2,4, and 8 have positive charge reorganizations of 0.80,0.66,0.73, and 0.69 electron, respectively;

-

-

o

+

3 p 331' ~ (7)

I

R', II

+

3 6 ~ 2431' ( 8 )

I *

3622 531' ( 9 )

Figure 9. Charge density difference plots for the first nine vertical triplet excited states of acetaldehyde at theCIS/d3 11(2+)G* level with respect to the ground state. The outermost contour is 0.0001 e/B3.

however, states 3, 5, 6, 7, and 9 have corresponding values of 1.06, 1.04, 1.13, 1.13 and 1.15 electrons, respectively. The Rydberg states, therefore, resemble a single excited electron in



TABLE X V Geometrical Parameters for the 3A" Triplet State of Acetaldehyde (6-31+6*)* parameter r(C-0) r(C-C) r(C-H,) r(C-Hb) r(C-Hc) r(C-Hd) L(0-c-C) L(C-C-H,) L(C-C-Hb) L(C-C-HJ L(C-C-Hd) T ( Ha-C-O-C) T ( H b-C-C-0) T ( Hc-C-C-0) T ( Hd-C-C-0) NIMAG V

energy'

Cs(eel) 1.2612 1 S204 1.0797 1 .OS22 1.0854 1.0854 120.94 123.18 109.52 111.13 111.13 180 0 119.48 -1 19.48 2 554.51 111.81 0.770 06 4.1 1

CIS Cs(stag) 1.2587 1.5202 1.0792 1.0821 1 .OS57 1.0857 120.37 123.20 107.83 1 1 1.71 1 1 1.71 180 180 60.87 -60.87 1 607.81

1.2539 1.5438 1.097 1 1.0817 1.0816 1 .OS66 116.76 113.10 109.59 107.99 111.81 131.94 -57.80 63.40 -176.25 0 201.8

0.771 21

0.776 28

CI

Cs(eel) 1.3479 1.4954 1.0712 1 .OS43 1 .OS69 1 .OS69 117.83 127.52 110.03 111.70 111.70 180 0 119.59 -1 19.59 2 614.11 144.61 0.835 32 2.33

UHF Cs(stag) 1.3467 1.4914 1.0707 1.0833 1.0877 1.0877 117.51 127.80 109.36 1 1 1.76 1 1 1.76 180 180 60.38 -60.38 1 649.51

CI

UMPZ

CI

1.3452 1.5023 1.0794 1 .OS45 1.0835 1.0886 114.56 119.80 110.83 109.55 111.68 139.65 -54.61 65.98 -173.88 0 204.5

1.3376 1.5047 1.0935 1.0917 1.0924 1.0975 114.23 119.07 108.97 110.49 111.80 137.83 -55.94 64.80 -174.90

0.836 92

0.841 24

1.232 69 3.75p

4.08 3.94 transn energyd 2.29 2.17 All bond distances in angstroms and angles in degrees. * Vibrational frequencies were not calculated. Absolute energy is -(152 In electronvolts. e The corresponding MP2/6-31+G* energy for the ground state is -153.370 48 hartrees.

a large atomic orbital around a radical cation. We do note that, like the singlet excited states, there is some mixing of the valence and Rydberg states that possess the same overall symmetry but arise from different occupied molecular orbitals (for example, state 8). Thecharge reorganizationofthe 3(n-r*) statesuggests that, as in the singlet excited state, it will be pyramidally distorted in its adiabatic geometry. Adiabatic Triplet Excited States

-

We were interested in examiningthelargechangein the charge density distribution for the triplet n a* state. Using the 6-31+G* basis set, we have calculated the optimized geometry of this state using the CIS, UHF, and UMPZ methods. The geometric information is listed in Table XV. This state has also been studied e~perimentally.~~ Both the CIS and UHF methods provide the same qualitative information for the geometrical distortions of the triplet n r* state. We will discuss the CIS method first. Optimization under C, symmetry for the eclipsed orientation of the methyl group afforded a structure with an elongated C-0 bond length (1.26 A) and changes in the bond angles around the carbonyl carbon. This structure is a second-ordersaddle point where one imaginary vibrational frequency correspondsto methyl rotation and the other to pyramidalizationof the carbonyl carbon. Indeed, the staggered orientation of the methyl group (also C, symmetric) is a true transition state. Optimization under C1 symmetry yielded a pyramidalized geometry around the carbonyl carbon and the C-H bonds of the methyl group remaining staggered to the C-O bond. The carbonyl carbon is pyramidalized by -48' (r(Ha-C-0-C) = 132O), and this is in good accord with the experimental value of 4So.55b In addition, it is known experimentally that the triplet n r* state is more pyramidalized than the singlet state,55band the CIS geometries also reproduce this feature. The C-0 bond length is essentially the same as that the C, structure, but the C-C bond length increases to 1.54 A. The bond angles around thecarbonylcarbonalsodecreaseto 117O and 113O forthe0-C-C and C-C-Ha angles, respectively, in the C1 structure. The UHF method provides similar qualitative behavior, but the C-O bond length is increased to 1.35 A and the bond angles change more drastically. The geometrical parameters of the CIstructure at the UMPZ level are between those of the CIS and UHF methods, but more similar to thoseof UHF; however, thecalculated UMP2 transition energy is more similar to thevalue from the CIS method. In particular, the adiabatic transition energy is 3.94, 2.17, and

-

-

b b

+ value) hartrees.

3.75 eV for CIS, UHF, and UMPZ levels, respectively (3.38 eV, experimental) .55b The vibrational frequencies of the triplet n r* state also pose a good test of the CIS method. In Table XVI we have listed the calculated vibrational frequencies for this state as well as the scaled frequencies using the average scaling factors (Table XI) for the acetaldehyde isotopomers. The experimental data are also listed.55bWe note that similar to the other excited states of acetaldehyde, there is a significant amount of mode mixing for the C, and C1 geometries of the 3(n r*) state. Once again, we have utilized a normal coordinate analysis to assign the vibrational frequencies of the excited state in terms of those of the ground state. (The matrix products are available as supplementarymaterial.) The low-energy vibrationalfrequencies, in particular u5-u9 and u12-u15, are extensively mixed. The data in Table XVI do show that the CIS method provides good agreement with the experimental vibrational frequencies for both hd and d4 isotopomers, and the calculated (scaled) frequencies are within 60 cm-I of the experimental values.55b

-

-

Atomic Charges and Covalent Bond Orders As for formaldehyde, we have examined the atomic contributions to the molecular properties according to Bader's theory of atoms in molecules33as well as the covalent bond orders of Cioslowski and M i ~ o n The . ~ ~calculated atomic populations for the different vertical states of acetaldehyde are listed in Table XVII, and the covalent bond orders are shown in Table XVIII. All integrated populations were derived from the respective6-31 1(2+)G* wave functions using the ground-state MP2/6-31GS geometry for thevertical states and the CIS/6-3 1+G* geometries for the adiabatic states. Theground state of acetaldehyde shows that thecarbonyl bond is polarized C+-0-, with the carbonyl carbon bearing an atomic charge of +1.09 e, while the oxygen has a charge of -1.24 e. The hydrogen atom (Hb) eclipsed to the carbonyl group is more positively charged than the other hydrogens, demonstrating the polarization of the eclipsed C-H bond of the methyl group by the C-0 dipole. This effect has been noted previou~ly.~~ The methyl carbon atom shows a slight negative charge. These atomic populations show that there is a mechanism for electrostatic stabilization in the ground state.* Analysis of the staggered orientation of the methyl group with respect to the carbonyl bond reinforces this hypothesis. In the staggered geometry for the ground state, the C-0 bond is also


TABLE XM: Vibrational Frequencies. of the Acetaldehyde 13A“ 3(n

mode A’

scaled

3298 3195 3309 1714 1630 1288 1536 1087 993 419 3243 1613 1115 5541 112i

3319 3199 3055 1660 1616 1240 1497 1012 924 375 3283 1612 1128 670 202

3024 2914 2630 1441 1443 1107 1337 904 825 335 299 1 1440 1007 598 190

V5

v6 Vl

V8 v9

VI0 VI I

A”

VI2 VI3 VI4

VI5

Excited State (CIS/6-31+G*) CDlCDO

CI

v2 v3 v4

T*)

CHjCHO

Cs(eel)

VI

-

Hadad et al.

exptlb

C, (eel)

CI

scaled

exptlb

905

2445 2294 243 1 1694 1180 995 1194 950 782 385 2406 1165 848 4641’ 81i

2462 2298 2233 1627 1178 945 1158 741 882 349 2430 1168 892 520 153

2270 21 19 1963 1422 1059 850 1041 666 792 339 2240 1050 802 467 141

727

364

518 179

340 999 442‘ 137

-

In inverse centimeters. The average scaling factors between the experimental and R H F frequencies for the ground state are used to obtain the scaled frequencies for each state (see text). Reference 55b. Determined as the difference between the average of the O+,O-) (1+,1-), inversion doublet bands of VI^.

TABLE XMI: Acetaldehyde Atomic Populationsa 0

C(=O) state

R

total

a

1A’ GS (ecl) 1A’ GS (stag) 2A‘ n RC n a* (1) n 3s (2 (ecl)) n 3s (2 (stag)) n 3PY (3) n 3pX(4) T R* (6) 47, 4 8 , 410 I* (7) n 3d,! (8)

0.457 0.452 0.345 1.041 0.333 0.325 0.386 0.425 0.575 1.083 0.361

4.913 4.91 1 5.196 5.468 5.204 5.195 5.098 5.130 5.182 5.488 5.127

1.576 1.580 1.762 1.909 1.757 1.760 1.699 1.652 1.487 1.853 1.734

n

R*

(C,) a* (Ci)

1.052

1.899

3s (Cs, (ecl))

0.362

5.524 5.488 5.037

n n

---- --

1.670

Hb

H, total

a

(a) Vertic:al 9.241 0.955 9.242 0.976 8.628 1.032 8.843 0.976 9.160 1.041 9.151 1.046 8.973 1.024 9.209 1.034 8.943 1.013 8.744 0.974 9.100 1.047 8.791 8.805 9.226

HC

total

a

total

R

total

R

total

States 6.018 6.023 5.948 5.924 6.03 1 6.039 6.065 6.051 6.077 5.987 6.184

0.012 0.012 0.004 0.035 0.003 0.003 0.009 0.01 1 0.025 0.035 0.005

0.961 0.957 0.708 0.861 0.926 0.939 0.988 0.896 0.863 0.907 1.015

0.019 0.022 0.017 0.020 0.016 0.017 0.015 0.018 0.0 19 0.020 0.016

0.944 0.982 0.860 0.954 0.878 0.926 1.105 0.887 0.919 0.937 0.885

0.490 0.479 0.420 0.509 0.425 0.425 0.435 0.43 1 0.445 0.513 0.4 19

0.962 0.943 0.828 0.972 0.901 0.873 0.886 0.909 1.008 0.968 0.850

0.036

0.870 0.880 0.931

0.021

0.961 0.969b 0.877

0.510

0.974 0.96g6 0.935

(b) Adiabatic States 0.975 5.917 5.924 1.047 6.057

0.004

0.015

0.437

Using the 6-3 11(2)+G* basis set and in units of electrons. The vertical states are with the ground-state MP2/6-3 lG* geometry, while the adiabatic states use the respective CIS/6-3 1+G* optimized geometry. The wave functions are R H F for the GS (ground state), UHF for the RC (radical cation), and CIS for the singlet excited states (ES). Unless noted otherwise, the eclipsed conformation is implicit. Integrated electron sums for each state are the following: (a) vertical, G S (ecl), 24.002; G S (stag), 24.000; 2A‘ RC, 22.996; ES1, 23.993; ES2 (ecl), 24.000; ES2 (stag), 23.996; ES3, 24.001; ES4,23.990; ES6,24.001; ES7, 23.998; and ES8,24.012 electrons; (b) adiabatic, ES1 (Cs), 24.010; ESl (Cl), 24.004; and ES2 (CJ, 23.996 electrons. Average value for all methyl hydrogens.

strongly polarized, and the methyl carbon atom actually has a larger negative charge. In both the staggered and the eclipsed conformers, the hydrogen atom(s) syn to the 0 atom is more positively charged than the hydrogen atom(s) which is anti to 0. Thus, the eclipsed C-H bond dipoles are polarized C--H+ and aligned oppositeto the C+-0- bond dipole, thereby allowing some electrostatic stabilization. Since the staggered hydrogens syn to the C-O bond are not directly aligned, the extra charge on the methyl carbon will cause a larger C--H+ bond dipole and therefore willcompensate for the poor alignment. Thecovalent bond orders show thesame trends. Theanti C-H bond shows a larger covalent bond order in the staggered structure as the bond polarity is decreased. The covalent bond orders for the other methyl C-H bonds aresmaller in the staggered structure, but themethylcarbon charge is greater and the ionic contribution must be larger. The stabilizationof the eclipsed orientation of acetaldehydeis therefore driven by a polarization effect in the neighboring methyl C-H bonds that serves to align the C-H dipoles in order to stabilize the large C-0 bond dipole of the carbonyl group. Analysis of the vertical 2A’ (n) radical cation shows that the total atomic charge decreases for all of the atoms except for the carbonyl carbon atom which increases by 0.28 e. In particular, the 0 and Ha lose 0.62 and 0.26 e, respectively, while the C(Me) and the other hydrogen atoms decrease by -0.10 e or less. This

accumulation of charge on the carbonyl carbon seems counterintuitive, but it seems to be a general trend for radical cations.57 The charge density difference plot between the vertical n radical cation and the ground state is shown in Figure 10, and one can see that there is a significant accumulation of charge at the carbonyl carbon and also in the region between the C-0 bond and the eclipsed C-H(Me) bond. However, one should note that the u and r systems oppose each other in the C-O bond, and even though the r charge on 0 increases,the total population decreases (see Table XVII). The carbonyl carbon actually loses r charge t o the 0, but the u withdrawal more than compensates. It is also interesting to note that the charge reorganization around the carbonyl carbon in the formaldehyde n, radical cation is nearly superimposable on that for the acetaldehyde n radical cation. This is also evident in the calculated atomic populations for both radical cations with respect to the ground state. For example, the 0 atoms undergo changes in the s and total populations of +0.18 and -0.62 e (HCHO) and +0.19 and -0.62 e (CH3CHO). The carbonyl carbon atom shows similar behavior: -0.16 and +0.20 e (HCHO) and -0.1 1 and +0.28 e (CH3CHO) for the rand total populations, respectively. The aldehydic H atom shows -0.01 and -0.29 e (HCHO) and -0.01 and -0.26 e (CH3CHO) changes on going to the n radical cation. The small difference in the calculated carbonyl carbon atomic



TABLE XMIk Acetaldehyde Covalent Bond Orders'

c-c

c-0 state

7r

total

n-

1'A' GS (ecl) 1 'A' GS (stag) 2A' n RC n-r*(l) n 3s (2 (ecl)) n 3s (2 (stag)) n 3pY (3) n 3pX(4) n7r* (6) 6 7 , us, 610 x* (7) n 3d,1(8)

0.624 0.623 0.402 0.262 0.453 0.450 0.409 0.345 0.178 0.255 0.438

1.388 1.385 1.129 1.002 1.304 1.297 1.199 1.142 0.947 0.802 1.255

0.062 0.059 0.1 14 0.081 0.096 0.091 0.095 0.091 0.081 0.082 0.102

n

0.256

0.989 0.988 1.376

0.077

n

n

--- -7r* 7r*

(C,)

(Cl)

3s (C, (ecl))

0.5 12

0.07 1

C-Ha

C-Hb

C-H,

total A (a) Vertical States 1.033 0.004 1.026 0.005 0.971 0.001 0.937 0.032 0.877 0.001 0.868 0.001 0.001 0.892 0.917 0.002 0.010 0.953 1.026 0.033 0.887 0.001

total

7r

total

7r

total

0.942 0.944 0.820 0.846 0.750 0.751 0.741 0.774 0.856 0.897 0.749

0.018 0.021 0.017 0.018 0.0 17 0.018 0.015 0.018 0.017 0.0 17 0.0 16

0.97 1 0.986 0.941 0.972 0.941 0.952 0.945 0.95 1 0.948 0.949 0.943

0.9 15 0.914 0.858 0.923 0.881 0.884 0.880 0.872 0.860 0.927 0.871

0.968 0.964 0.909 0.967 0.924 0.920 0.924 0.923 0.926 0.95 1 0.916

(b) Adiabatic States 0.944 0.033 0.918 0.001 0.743

0.878 0.855 0.711

0.019

0.972 O.97lb 0.944

0.926

0.971 0.97 1 0.939

0.016

0.910

Using the 6-311(2+)G* basis set. The wave functions are R H F for the GS (ground state), UHF for the RC (radical cation), and CIS for the singlet excited states (ES). The vertical states use the ground-state MP2/6-31G* geometry, while the adiabatic states use the respective CIS/6-3 1+G* optimized geometry. Unless noted otherwise, the eclipsed conformation is implicit. Average value for all methyl C-H bonds.

-

-

significant mixing of the A A* state with some of the n 3p Rydberg states, the values are not identical, but they are similar. In general, the total population on 0 decreasesonly slightly (-0.1 e) as compared to the ground state, and the carbonyl carbon population increases. The strong polarization of the C-0 bond is therefore maintained. Each Rydberg state does have different total atomic populations though, as each state distributes the charge density into different spatial regions (cf. Figure 6). The covalent bond orders, on the other hand, are not sensitive to the diffuse charge density, and the Rydberg states are very similar to the radical cation for the A and total bond orders. One can also predict trends in geometries as well. For example, the C-0 bond order decreases only slightly for the n 3s Rydberg state, but the C-Ha bond decreases significantly to 0.75 from 0.94 in the ground state. The adiabatic geometry should therefore cause a slight change in the C-0 bond length and a more dramatic change in C-Ha. This is observed. It is also interesting to examine why then 3s Rydberg state remains eclipsed in its adiabatic geometry. The C-O bond is still highly polarized C+-0-, and examination of the atomic charges shows that the H atom eclipsed to the carbonyl is more positively charged in the n 3s Rydberg state than in the ground state. The methyl carbon atom also has an increased negative charge. This would generate a larger C--H+ bond dipole that could better stabilize the C-O dipole of the carbonyl group. These effects are also seen in the charge analysis of the vertical, staggered n 3s state. Thus, the larger charge separation in the methyl C-H bonds would provide a stronger stabilization of the C-0 bond via the alignment of the C-H bond dipoles, and the n 3s Rydberg state should have a larger barrier to methyl rotation than the ground state. This is observed e~perimentally.~' The bond orders also show that the C, eclipsed structure should be maintained as the C-Hb bond order in the n 3s Rydberg state is similar to that in the ground state but the ionic character is greater due to the larger charge separation. Analysis of the charge density for the adiabatic n 3s Rydberg state (Tables XVII and XVIII) shows that the C-0 bond is even more polarized than in the vertical state, and the C-H bond dipoles would exert a stronger electrostatic influence on the C-0 bond. The valence states show different properties. The n A*, (predominantly) A A*, and 6 7 , bg, u10 A* states (1, 6, and 7) show a largeincrease in charge for thecarbonyl carbon (+0.55, +0.27, and +OS8 e, respectively) and a concomitant decrease for the 0 (-0.40, -0.30, and -0.50 e, respectively). The C-O bond dipole, therefore, is decreased significantly. Ha is affected more strongly than the other hydrogens ( -0.10 e), and in fact, there is almost no change for Hb and for H, in any of the valence states

-

Figure 10. Charge density difference plot for the vertical n radical cation with respect to the ground state for the molecular plane (6-31 1(2+)G*). The molecule is drawn to scale, and the oxygen atom is to the right. The out-of-plane (methyl) C-H bondsaredashed. Theouter contour is 0.0001 e/B3, and the contours increase by a factor of 2.

populations between formaldehyde and acetaldehyde must there fore be due to the charge distribution between the C-0 bond and the eclipsed C-H(Me) bond in the latter compound. The covalent bond orders show the same trends as the atomic populations for the acetaldehyderadical cation. The C-O covalent bond order decreases by -0.26 in the vertical radical cation, and 0.22 of that decrease is loss of A bond order. The C-C bond suffers only a small change in total bond order and actually increases the A contribution. All of the C-H bonds become weaker, but the C-Hb, the eclipsed C-H(Me) bond, decreases the least. The stabilization of the carbonyl group by the C-H eclipsed bond is still present, but it is dampened. Since the ionic character of the C-O bond is also smaller in the radical cation, all of these results would suggest that the C-0 bond is weakened in the radical cation, but the methyl stabilization of the carbonyl group is maintained. Therefore, one would predict that the C-O bond would lengthen in the radical cation, but the C, eclipsed structure would be maintained. This is observed experimentally and in our calculations. Let us now consider thevertical excited states of acetaldehyde. The predominantly n Rydberg states (2, 3, 4, and 8) show properties similar to those of then radical cation, especially with regard to the A atomic charges and bond orders. Since there is

-

-

-

-

-

-

-

-

N

-

Hadad et al.

4310 The Journal of Physical Chemistry, Vol. 97, No. 17, 199.3

states, especially with regard to the preference for planarity or pyramidalization of the carbonyl carbon in the different excited states. However, the C-0 bond length is underestimated at the CIS level, and this is similar to the known deficiency of HartreeFock theory to describe strongly polarized bonds, such as C-O bonds.24For these situations,further electron correlation is needed to predict the C-0 bond length adequately, such as CIS-MP2.9 The CIS method can also provide vibrational frequencies for the excited states, and the calculated frequencieshave to be scaled by the normal scaling factors of the ground state (-0.9).'ci2,29 The derived vibrational frequencies for the adiabatic geometries provide good agreement with experiment, and once again, the CIS method predicts the correct relative trends between excited states. The C-0 stretching frequency is calculated to be too high as the C-0 bond length is underestimated; however, the calculatedvibrational frequencies aregenerally within -60 cm-I of the experimental values for the different states. The CIS method may therefore provide a consistent starting point for deriving experimental force fields for the different excited states of carbonyl compounds. In particular, for a molecule such as acetaldehyde, where there is significant mode mixing in the different excited states, a theoretically derived set of force constants may be essential. The CIS method is also able to provide good agreement for both singlet and triplet states, and the calculated transition energies, adiabatic geometries, and frequency analyses of the triplet states of formaldehyde and acetaldehyde yield good agreement with experiment. The charge density analysis also reveals the greater degree of compactness of the triplet valence states as compared to the Rydberg states. Finally, the analysis of the charge density distributions into atomic contributionsto molecular properties,accordingto Bader's theory of atoms in molecules,33as well as covalent bond orders34 was very insightful. The atomic populations showed that there is a large degree of charge transfer to carbon in all of the excited Conclusions states studied, including valence, Rydberg, and radical cation states. A large portion of the transferred charge is from the H We have shown that the CIS method reproduces the known atoms. The 0 atom loses charge in the valence excited states, experimental trends for the excited states of formaldehyde and but in most cases, there is only a small charge loss from the 0 acetaldehyde. Our analysis of the wave functions for each state atom in the Rydberg states. There is charge flow in both the u revealed an unambiguous assignment of the nature of each and A systems of the excited states, and moreover, the u and ?r excitation, and furthermore, the use of charge density difference systems oppose each other. For instance, in the vertical n, T* plots showed the unique nature of the valence states and the state of formaldehyde the 0 atom actually gains 0.34 e in the A atomic-orbital-like character of the Rydberg states. Diffuse system (as compared to the ground state), but loses 0.38 e overall. functions are very important for an accurate description of the Thus, there is an overall 0.72 e charge transfer from 0 to C in excited states, even for some of the valence states which are mixed the u system. This opposition of u and A seems to be a general with Rydberg states of the same symmetry. The relative feature of charge density distributions.2 experimental ordering of the vertical transition energies is well reproduced at the CIS level if a flexible basis set is used, for The charge density analysis showed that the n A* and A example, 6-31 1(2+)G* or 6-31 1(2+,2+)G**. A* states have significantly decreased A covalent bond orders as compared to the ground state (see for instance, Table XVIII). The CIS method provides good agreement with the experiThese states are observed to pyramidalize in their adiabatic mental transition energiesfor thevalence states: but the excitation geometries. The preference for retaining an s-cis (eclipsed) methyl energies are high by 1.5 eV for the Rydberg states. A further C-H bond to the carbonyl group in the radical cation and Rydberg inclusion of electron correlation at the MP2 level (CIS-MP2) states of acetaldehyde is due to the favorable alignment of the provides better agreement with experiment, typically within 0.4 eV for the Rydberg states. Agreement for the valence states is strongly polarized C+-C- bond dipole with the corresponding within 1 eV at both the CIS and CIS-MP2 levels, and the transition bond dipole of the eclipsed C--H+ bond. The larger rotational energies at the CIS-MP2 level for these states tend to be higher barrier in the n 3s Rydberg state could be easily rationalized than the experimental values. Both the CIS and CIS-MP2 by the larger C-H bond dipole as compared to the ground state. methods, though, provide the correct relative energetic trends, as The valence states have a much smaller C-0 bond dipole due to compared to experiment. The CIS method also provides addithe large charge transfer to the carbonyl carbon, and the C-H tional information on states which have still not been assigned, bond dipole stabilization is less important. Rehybridization, via or determined conclusively, for instance, the A A* transition. pyramidalization of the carbonyl carbon, would account for the In agreement with suggestionsby Kupperman and co-worker~,~~ C-O covalent bond orders of 1 in the adiabatic valence states. the A A* transition energy is essentially the same for both Thus, there is a decreased stabilization of the carbonyl bond by formaldehyde and acetaldehyde (-9.1 eV). the methyl group in the n A* state as compared to the ground state, and this effect would explain the observed increase in the The CIS method yields adiabatic geometries of the excited vertical transition energy of the n A* state with increasing states that are quite similar to the experimental geometries. Good number of methyl substituents attached to the carbonyl group.4 agreement was found for the geometries of valence and Rydberg

as compared to the ground states. The methyl carbon atomic chargechanges by less thanO.l ein thevalencestates, andactually decreases in states 1 and 7, but increases in state 6. The covalent bond orders show the similar trends. The C-O bond order decreases for all three valence states, and the A bond order is drastically affected. There is actually an increased u covalent bond order for the C-0 bond as compared to the ground state. This effect must be related to the decreased polarity of the C-O bond. There is also a large change in the C-H, covalent bond order in all of the valence excited states. This change as well as the dramatic decrease in the ?r C-0 bond order must thereforeaffect the preference for pyramidalization at thecarbonyl carbon in the valence states. Pyramidalization and the corresponding rehybridization of the carbonyl carbon would better account for the C-0 bond order of -1 or less in the valence states. Indeed, analysis of the charge density distributions for the adiabatic n A* geometries (C,and C,) shows that the values for the atomic charges and covalent bond orders in the vertical states are maintained in the adiabatic geometries. The decreased polarity of the C-0 bond and the positive charge on the methyl carbon atom would disfavor any electrostatic stabilization of the C-0 bond by the methyl C-H bonds. Pyramidalization of the carbonyl carbon would therefore be expected, and this is observed. In summary, all of the excited states show significant charge transfer to the carbonyl carbon atom at the expense of the 0 and the H atoms. The Rydberg excited states show behavior similar to that of the radical cation, but the valence states are unique in their properties. The atomic properties and covalent bond orders from thevertical wave functions provide a method of rationalizing and/or predicting the geometric distortions in the excited states. Finally, some of the polarization of the eclipsed C-H(Me) bond by the carbonyl group is maintained in the Rydberg and radical cation states.

-

-

-

-

-

-

-

-

-

-

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Excited States of Carbonyl Compounds We have therefore shown that the CIS method can be very useful for examining the vertical and adiabatic geometries of the excited states of formaldehyde and acetaldehyde. The method, in conjunction with an analysis of the charge density, provides a powerful tool for examining the preferences of these molecules in their electronically excited states. Our analysis of the excited states of acetone will be presented subsequently.

Calculations All ab initio calculations reported here were performed with the GAUSSIAN 92 series of programs.58 Standard basis sets were used in all cases, except for 6-311(2+)G*, which is the 6-311+G* basis set with an additional sp shell on carbon (0.0131 928 exponent) and on oxygen (0.0254 518 exponent), and 6-31 1(2+,2+)G**, which is the 6-31 l++G** basis set with the above additions and also an additional s shell on hydrogen (0.0108 434exponent). Sixd Cartestianfunctions (the6D option) were used for all basis sets. The CIS and MP2 calculations correlated all electrons (i.e., full). All optimizations utilized the analytical gradient method. CIS frequency analyses were performed via numerical differentiation of the analytical gradient. The relative energies listed in the text do not include the correction for the zero point energy so as to render easy comparison of vertical and adiabatic energies. The oscillator strengths were calculated from the one-particle density matrix, while the wave functions were derived from the generalized density matrix9J9 and converted to natural orbitals with PSICHK.60 Charge density difference plots were obtained by calculating a cube of charge density (CH20,30 au on a side, and CHjCHO, 42 au on a side) with 80 points in each direction. Contouring and plotting were done using a modification of the program of J0rgen~en.l~ Atomic properties were calculated with the Atoms in Molecules Package (AIMPAC) from McMaster University and, in particular, a modified version of PROAIM61 and PROMEGA.62 Covalent bond orders were calculated with BONDER.2x34The vibrational frequency analyses were carried out by programs written by Demp~ey,~’ which are modified versions of those written by Schachtschneider.63

Acknowledgment. This research was supported by a grant from the National Science Foundation. We are also indebted to the Pittsburgh Supercomputing Center for a generous grant of computer time. C.M.H. graciously thanks the Fannie and John Hertz Foundation for a predoctoral fellowship. We thank Professor Lionel Goodman and his group (Rutgers University) for many insightful discussions as well as for providing some experimental information on acetaldehyde prior to publication. We also thank Todd Keith and Jim Cheeseman (McMaster University) for helpful discussionsand for providing PROMEGA prior to publication. Supplementary Material Available: Tables of calculated vibrational frequencies for the deuterated isotopomers of the ground state of acetaldehyde as well as the degree of vibrational mode mixing for the isotopomers of the ground, n A*, A , A*, n 3s, and n radical cation states of acetaldehyde (20 pages). Ordering information is given on any current masthead page.

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Excited states of carbonyl compounds. 1. Formaldehyde and

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