1498
J. Phys. Chem. 1996, 100, 1498-1504
Vibrational Stark Spectroscopy. 2. Application to the CN Stretch in HCN and Acetonitrile J. R. Reimers,*,† J. Zeng,† and N. S. Hush†,‡ Department of Physical and Theoretical Chemistry and Department of Biochemistry, UniVersity of Sydney, NSW 2006, Australia ReceiVed: August 16, 1995; In Final Form: October 16, 1995X
Finite-field ab initio calculations at the MP4 and/or QCISD levels are performed on HCN and acetonitrile (CH3CN) to calculate molecular dipole moments, polarizabilities, and hyperpolarizabilities as a function of the CN molecular vibration; these are then used to simulate the electroabsorption responses of these molecules, determining, for example, the Stark tuning rate. Our theory for these responses (part 1 of this series) assumes an experimental situation in which the chromophores are isolated and isotropic, being held firmly in a lowtemperature glass. For simplicity, we also exclude anharmonic intermode couplings and neglect contributions from all but the largest tensor component of the molecular polarizabilities, etc. This work is inspired by the recent electroabsorption measurements of Chattopadhyay and Boxer on the CN stretch in p-anisonitrile. We find that, for CN stretch vibrations, solvent effects dominate the observed electroabsorption responses, and we are able to qualitatively interpret the observed spectra.
1. Introduction When an isolated molecule is subject to an applied electric field F, its vibrational energies can each be expanded as a Taylor series in the applied field as
h∆ν ) hν(F) - hν(0) ) -∆µ·F - 1/2F·∆r·F
(1)
where h is Planck’s constant and the coefficients are usually termed the Vibrational dipole moment change ∆µ and Vibrational polarizability change ∆r. This language is misleading, however, as these are not exactly equivalent to the changes in the expectation values of the dipole and polarizability operators, respectively, induced by the vibrational transition; this is because the very application of the electric field actually modifies slightly the properties that it is intended to measure.1 The first term, -∆µ, is actually the Stark tuning rate: it is a quantity which has been measured for a wide variety of samples, for example, orientated CO physisorbed in zeolites, and it can be calculated a priori with reasonable accuracy.2-4 Recently, a new form of electroabsorption (Stark) spectroscopy has been developed by Boxer et al.5,6 and applied by Chattopadhyay and Boxer to study vibrational Stark effects for the CN stretch in p-anisonitrile7 and acetonitrile.8 In this experiment, samples are dissolved in a low-temperature glass in which the chromophore molecules are assumed to be isolated, randomly orientated, and rigidly held such that they cannot reorientate when an external field is applied. The random orientation ensures that no ensemble-averaged linear response to the applied electric field occurs; while eq 1 still applies at a microscopic level, the macroscopic response is given by Liptay’s equation9,10 as
d 1 (F) - (0) ) De (0) + Fe (0) + ν ν ν h dν ν F R(χ) d2 He 2 2 (0) (2) 2h dν ν 2
[
]
where F is the magnitude of the applied field within the glass * To whom correspondence should be addressed. † Department of Physical and Theoretical Chemistry. ‡ Department of Biochemistry. X Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-1498$12.00/0
at each molecular site, is the molar extinction coefficient as a function of frequency, De, Fe, and He are expansion parameters, and R(χ) is a sensitivity function,1,7 which for the experimental apparatus used has the value 1/5. The relationship between the internal field vector F and the applied external electric field vector Fapp is assumed to be
F ) fFapp
(3)
A reliable estimate for f is not yet generally available; however, it is likely5 to be in the range 1 e f e 1.3. This imposes a limit on the accuracy of derived quantities. Recent work by Creutz et al.,11 however, proposes an experimental method for determination of the proportionality constant which, if generally applicable, will remove this uncertainty. It is assumed that the shape of the absorption band remains invariant on application of the field. (This is most likely not to apply if the absorption corresponds to more than one vibrational absorption process.) The effect of the field is thus limited to changes of the transition absorption frequency and to modifications of the transition moment. For each molecule in the sample, the individual absorption frequency is taken to vary in accord with eq 1 while the transition moment vector is taken to vary according to
M(F) ) M + A·F + F·B·F
(4)
where M, A, and B are the field-free transition moment, transition polarizability (second rank), and transition hyperpolarizability (third rank) tensors, respectively. Assuming that the field-free tensor quantities ∆µ, ∆r, M, A, and B are dominated by just one component, we may replace them by scalar quantities ∆µ, ∆R, M, A, and B, respectively; Liptay’s theory then expresses the expansion parameters as
De )
A2 2B 2A∆µ ∆R + , Fe ) + , He ) (∆µ)2 2 M M 2 M
(5)
These coefficients represent contributions to the electroabsorption signal whose shapes are proportional to the absorption band contour (/ν) and its first and second frequency derivatives, respectively. The first term, De, has the shape of the unperturbed absorption envelope; as in the corresponding perturbation of © 1996 American Chemical Society
CN Stretch in HCN and Acetonitrile
J. Phys. Chem., Vol. 100, No. 5, 1996 1499
TABLE 1: MP2-Optimized Geometry of CH3CN (in au) and the Cartesian (δxi ) Qimi-1/2, Normalized Such That ∑iQi2 ) 1) MP2 CN Stretching Normal Coordinate (in 10-3 au) for HCNa MP2 geometry
CN Cartesian displacement
atom
x
y
z
x
y
z
C N C H H H
0.000 00 0.000 00 0.000 00 0.000 00 1.678 98 -1.678 98
0.000 00 0.000 00 0.000 00 1.938 71 -0.969 36 -0.969 36
0.518 04 2.740 26 -2.241 58 -2.946 85 -2.946 85 -2.946 85
0.000 00 0.000 00 0.000 00 0.000 00 -0.035 72 0.035 72
0.000 00 0.000 00 0.000 00 -0.041 25 0.020 62 0.020 62
-5.357 44 3.722 89 0.872 96 0.556 29 0.556 29 0.556 29
a The corresponding CN and CH bond lengths are 2.2176 and 2.0148 au, respectively, while the displacements are 8.6922 and 3.2123 × 10-3 au, respectively.
an electronic transition, this is referred to as the “constant” term and arises from contributions from the transition moment polarizability and hyperpolarizability. The second term in the perturbation, Fe, has the shape of the first derivative of the unperturbed absorption envelope; this arises from contributions from the polarizability change ∆R and a cross-term involving A and ∆µ. The third term, He, has the shape of the second derivative of the absorption spectrum and is simply proportional to (∆µ)2; it thus yields unambiguously the absolute value of ∆µ when the proportionality factor f is known. Unfortunately, no general unique solution for the other three molecular properties, ∆R, A, and B, is available from the remaining two experimental observables, De and Fe. Here, we calculate M, A, B, ∆µ, and ∆R and hence the electroabsorption responses De, Fe, and He for HCN and acetonitrile. Finite-field quantum-chemical techniques are used as pioneered by Hush and co-workers.1,2,4,4,12-20 These are very useful for electroabsorption modeling as the application of the electric field modifies the molecular geometry and hence changes the molecular properties that it is designed to measure; finite-field techniques allow for a direct simulation of the real physical situation. The general theory and available options have been described in detail elsewhere in part 1 of this series1 where an application to the CO stretch is developed; other authors have also considered the basic theory, and this work has recently been reviewed by Martı´ and Bishop.21 Here, atomic units (au) are used throughout; conversion factors to other commonly used units include the following:
1 au ) 0.529177 Å (length r) 5.48593 × 104 amu (vibrational effective mass µv) 27.2107 eV (energy V) 219477 cm-1 (wavenumber ν) 5142.57 MV/cm (electric field F) 42.6784 cm-1 (MV/cm)-1 (Stark tuning rate) 2.54156 D (dipole moment µ,m) 0.148185 Å3 (polarizability R) 8.65710 × 10-33 esu (hyperpolarizability β) 2. Calculation Method Ab initio calculations are performed using GAUSSIAN-92/ DFT22 for HCN and CH3CN using an extensive basis set of the type especially developed and tested for its applicability to this type of problem.23,24 It is based on Dunning’s25 (9,5) f [5,3] contraction for C and N and (4) f [3] for hydrogen, extended
by diffuse s and p functions (ζs(N) ) 0.0711, ζp(N) ) 0.0551, ζs(C) ) 0.0511, ζp(C) ) 0.0382, and ζs(C) ) 0.0411) and two sets of 3d and 2p polarization functions (ζd(N) ) 0.80,0.15, ζd(C) ) 0.71,0.09, and ζp(H) ) 0.75, 0.15), resulting in a [6,4,2;4,2] basis (a total of 28 basis functions per heavy atom and 10 per hydrogen atom). For both molecules, the equilibrium geometry is optimized using second-order Møller-Plesset perturbation theory (MP2) and the CN stretching normal coordinate vector obtained; this results in calculated CN bond lengths of 2.2176 and 2.2222 au, harmonic frequencies of 2005.9 and 2198.2 cm-1, and vibrational effective masses µV of 5.7440 and 12.3955 amu (10 470 and 22 595 au), respectively. The calculated equilibrium geometries and the cartesian displacements corresponding to the CN normal coordinates are shown in Table 1. Displacements Q of the nuclear geometry away from the MP2 minimum along the MP2 CN normal coordinate are then applied, as is an external electric field F in the CN-bond direction, and the molecular potential energy V(Q;F) is evaluated on a grid of 7 × Q and 5 × F values. A variety of methods are employed for calculating these energies, including the self-consistent field (SCF), MP2, MP4, and quadratically convergent singles and doubles configuration interaction (QCISD) methods. Calculated energies at the QCISD level are shown in Table 2 for HCN while energies at the MP4 level are shown in Table 3 for CH3CN. From these energies, the molecular dipole moment µ(Q;F), polarizability R(Q;F), and hyperpolarizability β(Q;F) are evaluated by numerical differentiation, and from these the electroabsorption properties are evaluated. 3. Calculated Electroabsorption Properties The simplest method1 for determining M, A, B, ∆µ, and ∆R is to perform a variational vibrational analysis for the transition energy and moment based on the V(Q;F) and µ(Q;F) data; a separate variational analysis is performed at each field strength F yielding directly ν(F) and M(F) which are then numerically differentiated using eqs 1 and 4 to yield the electroabsorption parameters. An alternate1 analytical approach exists, however, which relates the electroabsorption parameters back to fundamental molecular properties. These two approaches do differ in their treatment of high-order field components, but at the field strengths used herein such differences are negligible. Clearly, both approaches are approximate in that they neglect the effects of anharmonic (and possibly also harmonic) couplings between the mode of interest and other molecular vibrations. In the analytical approach, we express1 the field-free molecular potential energy V(Q), dipole moment µ(Q), polarizability R(Q), and first hyperpolarizability β(Q) as
1500 J. Phys. Chem., Vol. 100, No. 5, 1996
Reimers et al.
TABLE 2: Calculated QCISD Energy V+93 (in au) for HCN Displaced about Its MP2 Equilibrium Geometry by Q/Qzpta F
a
Q/Qzpt
0.04
0.02
0.00
-0.02
-0.04
-4 -2 -1 0 1 2 4
-0.069 486 884 -0.161 700 055 -0.178 754 097 -0.182 812 611 -0.177 317 808 -0.164 931 292 -0.127 239 790
-0.082 548 622 -0.173 333 098 -0.189 551 710 -0.192 697 078 -0.186 214 826 -0.172 770 925 -0.132 779 188
-0.102 557 854 -0.193 109 991 -0.209 141 009 -0.212 050 144 -0.205 284 005 -0.191 510 870 -0.150 741 823
-0.129 442 290 -0.220 842 612 -0.237 270 260 -0.240 553 807 -0.234 139 356 -0.220 696 522 -0.180 539 829
-0.163 443 322 -0.256 732 837 -0.274 121 966 -0.278 371 398 -0.272 925 375 -0.260 454 154 -0.222 266 692
Qzpt ) 10.4602 au is the MP2 zero-point displacement of the CN stretch, subject to a z field of magnitude F au.
TABLE 3: Calculated MP4 Energy V+132 (in au) for CH3CN Displaced about Its MP2 Equilibrium Geometry by Q/Qzpta F
a
Q/Qzpt
0.04
0.02
0.00
-0.02
-0.04
-4 -2 -1 0 1 2 4
-0.287 146 600 -0.380 896 370 -0.398 495 470 -0.402 553 660 -0.396 243 910 -0.381 972 280 -0.336 299 450
-0.297 371 120 -0.389 706 720 -0.406 541 590 -0.409 806 360 -0.402 674 610 -0.387 548 720 -0.340 030 660
-0.321 266 740 -0.413 242 530 -0.429 913 280 -0.433 036 850 -0.425 792 020 -0.410 582 640 -0.362 964 480
-0.358 597 440 -0.451 240 290 -0.468 340 260 -0.471 968 470 -0.465 308 660 -0.450 764 780 -0.404 678 420
-0.409 862 700 -0.504 312 170 -0.522 513 350 -0.527 383 180 -0.522 106 750 -0.509 082 150 -0.466 363 310
Qzpt ) 9.9922 au is the MP2 zero-point displacement of the CN stretch, subject to a z field of magnitude F au.
V(Q) ) V0 + a1(Q - Qe)2µV-1 + a2(Q - Qe)3µV-3/2 µ(Q) ) m0 + m1(Q - Qe)µV-1/2 + m2(Q - Qe)2µV-1 + m3(Q - Qe)3µV-3/2 (6) R(Q) ) R0 + R1(Q - Qe)µV-1/2 + R2(Q - Qe)2µV-1 + R3(Q - Qe)3µV-3/2 β3(Q - Qe)3µV-3/2 where Qe is the equilibrium value of the normal coordinate. (It may be nonzero as we use MP2 normal coordinates but evaluate the energy using different methods.) This equation is written in this form so as to be directly analogous with the standard treatment of a diatomic molecule4 and allows results for HCN and CH3CN to be directly compared as effects due to the differing reduced masses are removed (µV1/2 ) 102.32 and 150.32 au for HCN and CH3CN, respectively). In the presence of an applied electric field, the molecular energy and dipole moment thus become
V(Q;F) ) V(Q) - µ(Q)F - R(Q)F2/2 - β(Q)F3/6 µ(Q;F) ) µ(Q) + R(Q)F + β(Q)F /2
∆µH ) (hνH/4a12)(2m2a1 - 3m1a2)
(8)
Alternatively, first-order perturbation theory could be used1 with, for example, this revised approximate becoming
β(Q) ) β0 + β1(Q - Qe)µV-1/2 + β2(Q - Qe)2µV-1 +
2
included. Alternatively, a local-harmonic-oscillator analysis (i.e., the field-dependent minimum and curvature of a general anhamonic potential are found and analyzed1) may be performed for ν(F) and M(F), giving the electroabsorption responses directly. Full details have been given elsewhere with, for example, ∆µ being approximated by
(
∆µP ) ∆µH 1 +
)
15a22 15a2m3 hνH (hνH)2 3 4a1 8a13
(9)
We find that, for HCN and CH3CN, differences between quantities evaluated at the harmonic theory and perturbation theory levels differ by at most 10% and that perturbation theory and essentially exact variational results differ by at most 2%. The electroabsorption parameters M, A, B, ∆µ, and ∆R calculated using the variational analysis of the analytical function are also shown in Table 4, along with predicted values of De, Fe, He, and their components; note that this table actually shows values for the directly observable quantity M/γ, where
γ ) (hνH/4a1)1/2
(10)
(7)
Hence, we first determine the 16 molecular parameters present in eq 6 by collocating the calculated V(Q;F) data, and the results are shown in Table 4. It is difficult to obtain reliable values for high-order parameters such as m3, R2, R3, β2, and β3. A useful test for the reliability of these parameters is to perform the analyses using selected subsets of the available V(Q;F) data; we find that no parameter changes significantly when only five of the seven Q values are used, and only the values of the β’s change when four of the five F values are used. Several methods are available1 for determining M, A, B, ∆µ, and ∆R from the above 16 parameters. One method is to take eq 7 and solve it variationally as was previously described; this differs from the previous numerical solution in that no higherorder expressions in Q and F other than those specified are
At the highest levels of theory used we see that ∆µ is calculated to be very small, 0.0004 au for HCN and -0.0026 au for CH3CN. The variation between these two molecules originates from the change in sign of m2 shown in Table 4: for CH3CN, the two contributions in eq 8 add while they cancel for HCN. For both molecules, the first-derivative electroabsorption response Fe is dominated by ∆µA/M with the contribution form the polarizability change ∆R being much less. Similarly, (A/M)2 dominates the contributions from 2B/M to the constant term He. For HCN, Table 4 shows results obtained using the SCF, MP2, MP4, and QCISD methods, and V(Q), µ(Q), R(Q), and β(Q) are shown in Figure 1; for CH3CN, this table shows results obtained using the SCF, MP2, and MP4 methods, while Figure 2 shows the corresponding properties. We see that, for most
CN Stretch in HCN and Acetonitrile
J. Phys. Chem., Vol. 100, No. 5, 1996 1501
TABLE 4: Calculated and Observed Properties of HCN and CH3CN (in au) HCN property rCN νH νP νV a1 a2 m0 m1 m2 m3 R0 R1 R2 R3 β0 β1 β2 β3 M/γ ∆µ ∆R/2 A/M 2∆µA/M Fe 2B/Ml (A/M)2 l Del
SCF 2.133 2407 2371 2370 0.6294 -0.5967 -1.284 -0.208 -0.016 -0.019 21.5 16.1 4.2 -1 4.7 32 5 0 -0.2099 -0.0029 0.1267 -77.8 0.451 0.577 -0.11 6.1 6.0
MP2 2.218 2006 1959 1957 0.4372 -0.4768 -1.192 0.045 0.227 0.135 22.4 13.8 1.0 -2 0.4 28 -17 -21 0.0545 0.0036 0.1362 260 1.837 1.974 0.77 65.2 66.0
MP4 2.197 2115 2073 2071 0.4863 -0.5051 -1.196 -0.030 0.102 0.032 22.4 15.0 3.1 0 4.1 46 20 29 -0.0269 0.0006 0.1418 -560 -0.705 -0.564 -1.90 317 315
CH3CN QCISD
obs
2.195 2136 2096 2095 0.4959 -0.4991 -1.196 -0.033 0.085 -0.004 22.4 15.0 3.0 -1 8.4 53 5 -11 -0.0309 0.0004 0.1356 -490 -0.361 -0.226 -1.80 240 239
2.188j 2089 -1.174 -0.029a,b
-0.029a
SCF
MP2
MP4
2.072 2545 2515 2514 1.5195 -1.9717 -1.671 -0.404 -0.648 -0.657 37.3 25.9 21.2 4 -11.2 -40 -15 150 -0.4180 -0.0058 0.1491 -63.7 0.745 0.894 0.20 4.1 4.3
2.222 2198 2164 2164 1.1333 -1.5424 -1.547 -0.017 0.060 0.421 39.0 23.8 11.2 -3 -40.9 -140 -210 -34 -0.0132 0.0002 0.1402 -1800 -0.748 -0.607 11.10 3299 3310
2.181 2301 2268 2268 1.2419 -1.6619 -1.550 -0.142 -0.300 -0.142 38.6 24.5 14.5 -1 -30.3 -82 -50 320 -0.1476 -0.0026 0.1434 -170 0.891 1.034 0.67 28.4 29.1
obs 2.190j 2267c -1.542d,e (0.167b,k
38.7d,f
-34.7d,g
(0.167k (0.101/fh
1.4/f2h 20/f2hi
a Reference 47 (the sign is known19,48). b Obtained from M/γ assuming electrical and mechanical harmonicity. c The Fermi-resonance-corrected value41-43 is 2271 cm-1. d Latest treatment is ref 45. e Reference 49e. f Reference 50; the calculated44 CCSD(T) value is 38.7. g This is actually the CCSD(T) calculated result.44 h Determined by us from a preliminary spectrum of Chattopadhyay and Boxer8 in 2-methyltetrahydrofuran glass at 77 K. i Reference 51, ( 0.006 au; see also ref 52. i (20 × 103 au. j (0.006 au from ref 51; see also ref 52. k From ref 53 (1995) assuming the ideal gas law; older values are (0.13630 (1958), (0.179,54 and (0.12155 (1985). l In 1000 au.
of the key properties, the SCF values are quite poor while MP2 overestimates the required correction. For HCN, very similar results are obtained using MP4 and QCISD, and the results appear to be converging to within acceptable error limits as a function of the degree of configuration interaction included. Experimental results, where available, are also included in Table 4. The calculated CN bond lengths are within 0.01 au of the observed values at MP4 and higher levels, the variationally evaluated vibration frequencies νV differ by
