J . Phys. Chem. 1987, 91, 1749-1754
1749
elimination of H from POH. The reactive scattering and insertion pathways for H 2 H P O have been characterized via the IRC calculation and localized orbitals. The insertion barrier lies 6.9 kcal/mol above that for reactive scattering.
Office of Scientific Research (Grant 82-0190). The computer time made available by the North Dakota State University Computer Center is gratefully acknowledged.
Acknowledgment. This work was supported by the Air Force
Registry No. H3P0, 13840-40-9; H2, 1333-74-0; HPO, 13817-06-6.
+
Multipole Polarizabilities and Hyperpolarizabilities of AH, and A,H, Molecules from Derivative Hartree-Fock Theoryt Shi-yi Liu and Clifford E. Dykstra* Department of Chemistry, University of Illinois, Urbana, Illinois 61801 (Received: July 24, 1986; In Final Form: October 27, 1986)
Derivative Hartree-Fock (DHF) theory has been employed to analytically evaluate dipole and quadrupole polarizabilities and hyperpolarizabilities of simple first-row hydrides. A basis set augmentation for first-row elements that is moderately well suited for obtaining electrical properties and relatively economical is given. The isotropic dipole polarizabilities of AH, molecules are found to vary smoothly with the atomic number of the first-row element. This implies that the isotropic polarizabilities of hydrides are largely determined by the heavy atom, and DHF calculations carried out on the A2H, molecules, HCCH, H2CCH2,and N2H2,and HCN support that result. Second and third dipole hyperpolarizabilities,which are fourthand fifth-derivative properties, are also reported. Permanent first, second, and third moments were obtained at SCF and correlated levels. With the large basis sets used, the correlation effects are small in most cases.
Introduction Electrical polarization is the most important electronic structure change that occurs in molecules at long range, as two molecules just begin to approach each other. The long-range potentials1P2 used in molecular scattering studies can have much to do with collision and reaction cross sections and these are generally electrical in nature, involving only permanent and induced charge fields. The effects of fields at surfaces, field effects on vibrational spectra, and vibrational Raman and hyper-Raman Spectroscopy also relate to the polarization of a molecular charge di~tribution.~” In hydrogen bonding, electrical poIarization7~* is also of quantitative importance in the red shifts in the vibrational spectra arising from intramolecular s t r e t c h e ~ . ~ . ~ ~ Ab initio calculation of polarizabilities and hyperpolarizabilities has become increasingly tractable in recent years, making it possible to provide a strong theoretical basis for analyzing molecular electrical interactions. The polarizabilities are all second derivatives of the electronic energy with respect to field components, field gradient components, and so on. This means they can be calculated by finite differences. At the self-consistent field (SCF) level, they can also be found analytically by using coupled-perturbed Hartree-Fock Higher derivative properties, such as the second hyperpolarizabilities which are fourth derivatives, can be calculated analytically by the uniform, open-ended procedure of derivative Hartree-Fock (DHF) theory.I39l4 In fact, D H F ninth-derivative properties have been reported.I5 Derivative methods for correlated wave functions are also available (see, for example, ref 16-18, and references therein), though not yet to high order. We have used D H F to obtain multipole polarizabilities and hyperpolarizabilities of a number of first-row hydrides, and from these results we have identified certain trends. To provide sufficient properties for calculating intermolecular electrical interactions, we have calculated wellcorrelated, permanent moment values as well.
equation to any desired order and with respect to any number or any type of parameters in the Hamiltonian., For multipole polarizabilities, these parameters are the power series expansion terms of the electrical potential, e.g. the field components, the field gradient components, and so forth. The first detivatives of the Hamiltonian with respect to these parameters are the multipole moment component operators. For a distribution of charges, (qi), representative operators are
(1) Bernstein, R. B., Ed. Atom-Molecule Collision Theory: A Guide f o r the Experimentalist; Plenum: New York, 1979. (2) Abusalbi, N.; Eades, R. A,; Nam, T.; Thirumalai, D.; Dixon, D. A.; Truhlar, D. G.; Dupuis, M. J. Chem. Phys. 1983, 78, 1213. (3) King, F. W.; van Duyne, R. P.; Schatz, G. C. J . Chem. Phys. 1978, 69, 4472. (4) Brieger, M. Chem. Phys. 1984, 89, 275. (5) Andrews, D. L.; Sherborne, B. S. Chem. Phys. 1984, 88, 1. (6) Malik, D. J.; Dykstra, C. E. J . Chem. Phys. 1985, 83, 6307. (7) Liu, S.-Y.; Dykstra, C. E.; Malik, D. J. J . Mol. Structure 1986, 135, 357. (8) Liu, S.-Y.; Dykstra, C. E. Chem. Phys. 1986, 107, 343. (9) Liu, S.-Y.; Dykstra, C. E. J . Phys. Chem. 1986, 90, 3097. (10) Bernholdt, D. E.; Liu, S.-Y.; Dykstra, C. E. J . Chem. Phys. 1986,85, 5 120. (1 1) Stevens, R. M.; Pitzer, R. M.; Lipscomb, W. N. J . Chem. Phys. 1963, 38, 550. (12) Gerratt, J.; Mills, I. M. J . Chem. Phys. 1968, 49, 1719. (13) Dykstra, C. E.; Jasien, P. G. Chem. Phys. Lett. 1984, 109, 388. (14) Dykstra, C. E. J . Chem. Phys. 1985, 82, 4120. (15) Liu, S.-Y.; Dykstra, C. E.; Malik, D. J. Chem. Phys. Lett. 1986, 130, 403. (16) Handy, N. C.; Schaefer, H. F. J . Chem. Phys. 1984, 81, 5031. (17) Harrison, R. J.; Fitzgerald, G. B.; Laidig, W. D.; Bartlett, R. J. Chem. Phys. Lett. 1986, 124, 291. (18) King, H. F.; Komornicki, A. J . Chem. Phys. 1986, 84, 5645.
Theoretical Approach
Derivative Hartree-Fockl’ is a uniform method for solving the equations obtained by directly differentiating the Hartree-Fock The electrical properties reported here, or mentioned but not listed, are being incorporated into a data base of molecular electrical properties. For further information, contact the authors.
0022-3654 I87 I209 1- 1749SO1.50 I O ,
#
I
0 1987 American Chemical Societv -
1750 The Journal of Physical Chemistry, Vol. 91, No. 7, 1987
Liu and Dykstra TABLE I: Basis Set Augmentation to Atomic Triple-{ Sets: The ELP Bases
We will use the ”Cartesian” form of the moments and the moment operators rather than the traceless form.” In certain prior s t ~ d i e s , ~ we * ’ ~have * ~ ~not used the and factors in the diagonal moment operators and all polarizabilities involving those operators will differ from those following the convention used here by an appropriate factor of or ‘/6. (For instance, the dipole-quadrupole polarizability will have a ’/* factor while the quadrupole-quadrupole polarizability will have a (‘ / J 2 factor.) The moments, as opposed to the moment operators, are reported ...) because without using the presummation factors (e& that is most common. The interaction Hamiltonian is the power series or moment expansion of &iV(xi,yi,zi) where V is the electrical potential. Convergence in most of the D H F calculations typically required -20 iterations per derivative orbital set. For NH3, the usual process was divergent for certain derivatives. However, this was easily corrected by adding a small constant to the energy denominator of the iterative correction e q ~ a t i 0 n . lThis ~ provided a good initial guess of the derivative orbitals in just a few “damped” iterations and prevented divergence. The small constant was then removed and good convergence to the final result followed. There are two important concerns in assessing the reliability of the calculated properties, one of which is electron correlation. For covalent species such as those studied here, the refinements in dipole polarizabilities from including electron correlation effects are often 10-15%.2i-24 Of the species considered here, lithium hydride is probably furthest from being covalent, and so, the correlation effect is larger, about -20-30%.25J6 It is possible that the correlation effects could be greater for higher derivative properties, but a general assessment of that sort remains to be done. Basis set quality is very important in obtaining electrical properties. Molecular charge polarization can be both intraatomic polarization around constituent atoms and valence charge polarization, a shift of charge density along the molecular skeleton. The description of intraatomic polarization has the most stringent basis set requirements, with diffuse and high-I-type functions being very essential, just as they would be in describing polarization of an isolated atom. Interatomic valence polarization is not as demanding in basis set flexibility and is described partly with usual valence basis functions. It is clear that small molecules require the most carefully chosen basis sets. These sets must be quite extended14~2i~26-29 and this is because intraatomic polarization is important and cannot be adequately described with the functions of other centers. The last percent of the values in the approach to the basis set limit values may happen very slowly with basis set augmentation.28 The basis sets used here were based on those of several prior s t ~ d i e which s ~included ~ ~ basis ~ ~set tests ~ ~ to ensure that, for polarizabilities, the sets were moderately well converged. It is expected that the dipole polarizabilities are mostly within 5% of basis set limit values, while quadrupole polarizabilities and dipole hyperpolarizabilities have a greater error, possibly 20%. Comparisons that support this are given later along with selected basis set comparisons. The basis sets that were used here were
exponents of uncontracted Gaussians element
0.9 0.1
Li“
0.008
5.0 1.o 0.15 0.02
0.9 0.13 0.02
C
0.05
0.03
0.9 0.13 0.02
N
0.06
0.04 0.006
0.9 0.13 0.02
0
0.06
0.05 0.007
0.9 0.13 0.02
F
0.06
0.06 0.008
0.9 0.13 0.02
“For lithium, the augmentation was to a 5s valence set without p functions. TABLE 11: Dipole Polarizabilities and Hyperpolarizabilities of AH, Molecules (in au). LiH CH,b NH, H20 HF 16.00 a 23.45 12.82 8.410 4.751 22.32 24.02 24.02
16.00 16.00 16.00
13.19 12.64 12.64
8.313 9.077 7.839
5.612 4.320 4.320
-353.2 -137.6 -137.6 0.0 0.0
0.0 0.0 0.0 -11.03 0.0
9.126 7.802 7.802 0.0 -9.788‘
5.472 10.03 0.545 0.0 0.0
8.274 -0,100 -0,100 0.0 0.0
Y ~ , ~ . ~5 1598 . ~ 15 130 Y*,x.y,y Y ~ . ~ . ~ 151 . ~ 30 42280 YYYSY 14093 Yy,y,1,2 0.0 YX.YS,Y 42280 Yz,z,z,z
1679 613.0 613.0 1679.0 613.9 0.0 1679.0
4261 1055 1055 1070 356.6 -119.9 1070
794.0 288.6 343.4 475.2 300.1 0.0 1309
262.9 80.77 80.77 312.8 104.3 0.0 312.8
6,,X,,,x,X
0.0
18272 7954 7954 4622 4622 1540 -1794 -7270 1454 436.1 0.0
5266 2325 836.0 3405 519.7 1193.0 0.0 0.0 0.0 0.0 0.0
2309 243.5 243.5 91.06 91.06 30.35 0.0 0.0 0.0 0.0 0.0
aYy
az,z
fix,,,, Px,Yg P,.,,, PX,Y,i
PYYY
~
-5.225 X lo6
~ bx,x,x,y,Y ~ ~ -1.092 ~ ~ X~ lo6~ 0.0 ~ 6,,X,X,,,, 6x,y,Y,Y,Y 6,,,z,z,,
6x,Y,Y,i,2 6X,XY.Y.Y
6,,,,J’, ~YY.Y,Z,i ~Y,Z.Z,i,i
374. (27) Christiansen, P. A.; McCullough, E. A. Cfiem. Phys.Lett. 1978, 55, 439. ( 2 8 ) Bishop, D. M.; Maroulis, G. J. Cfiem. Pfiys. 1985, 82, 2380. (29) Roos, B. 0.;Sadlej, A. J. Cfiem. Pfiys. 1985, 94, 43. (30) Liu, S.-Y.; Dykstra, C. E.; Kolenbrander, K.; Lisy, J. M. J . Chem. Pfiys. 1986, 85, 2077.
d
P
0.06
0.005
-
(19) Buckingham, A. D. Q.Rev. Cfiem. SOC.London 1959, 13, 183. Adu. Cfiem. Pfiys. 1967, 12, 107. (20) Liu, S.-Y.;Dykstra, C. E. Cfiem. Pfiys.Lert. 1985, 119, 407. (21) Werner, H.-J.; Meyer, W. Mol. Pfiys. 1976, 31, 8 5 5 . (22) Bartlett, R. J.; Purvis, G. D. Pfiys.Rev. 1979, AZO, 1313. (23) Morrison, M. A.; Hay, P. J. J . Cfiem. Pfiys. 1979, 70, 4034. (24) Amos, R. D. Cfiem. Pfiys. Lett. 1980, 70,613. 1982,88, 89. Mol. Pfiys. 1980, 39, 1 . (25) Gready, J. E.; Bacskay, G. B.; Hush, N. S. Cfiem. Pfiys. 1977, 23, 9. 1977, 24, 333. (26) Karlstrom, G.; Roos, B. 0.;Sadlej, A. J. Chem. Pfiys. Lerr. 1982,86,
S
H
~x.x.x,y.z
-1.092 -1.363 -1.363 -4.542 0.0 0.0 0.0 0.0 0.0
X X X X
lo6 lo6 lo6 lo6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -3770
” T h e x axis is the symmetry axis for all molecules. The H’s are always located in the -x direction and the heavy centers are in the +x direction. Any values not listed are related by symmetry. bResults for a , P, and y were reported in ref 17. The ELP basis was “Basis A” in that report. cPY,z,l = -P,,Y, by symmetry in NH,.
built on Dunning-c~ntracted~~ H u ~ i n a g triple-{, a ~ ~ TZ, core-valence sets. In the case of lithium, the basis was built on a (10s/5s) contracted set, while for hydrogen it was a (6s/3s) contracted set. The basis sets were augmented with diffuse valence functions and are triply polarized, except for hydrogen which is doubly polarized. (31) Dunning, T. H. J . Cfiem. Pfiys. 1971, 55, 716 (32) Huzinaga, S. J. Chem. Phys. 1965, 42,1293.
Polarizabilities and Hyperpolarizabilities of AH, and A2H,
The Journal of Physical Chemistry, Vol. 91, No. 7 , 1987
1751
TABLE III: Dipole Polarizabilities and Hyperpolarizabilities of AIH, Molecules (in au)O & ax,x aYY %z PX,,,,
PXJ,
PX.Y,Y PX,,,,
PYYY PY,Z.Z
co
NZb
HCCHb
HCN
H2CCH2
trans-HNNH
12.13 14.2 1 0.0 11.09 1 1.09
11.38 14.83
23.15 31.36
16.63 22.40
0.0
0.0
0.0
19.04 19.04
13.75 13.75
18.18 23.91 2.984 17.38 13.25
0.0
9.66 9.66
28.08 36.77 0.0 24.69 22.78
30.8 1 0.0 5.026 5.026 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
7.346 0.0 2.255 2.255 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 -1 1.62 0.0 0.0 -17.95 6.652
cis-HNNH 18.53 25.03 17.01 13.56
“ T h e x axis is the axis connecting the two first-row elements. The y axis is in the molecular plane. For CO, the carbon is in the negative x direction relative to oxygen. From ref 14.
TABLE I V Isotropic Dipole Polarizabilities in au
co N2 HCN HCCH H2CCH2 trans-N2H2 cis-N2H2
calcd
est with eq 1
diff
12.13 11.38 16.63 23.15 28.08 18.18 18.53
12.41 13.64 16.82 20.00 26.00 19.64 19.64
2% 16% 1% -16%
-8% 7% 6%
TABLE V Dipole-Quadrupole and Quadrupole Polarizabilities (in au) LiH HF CO N2 HCCH HCN -44.11 PxYy 4.419 Py,xy -23.11 ,,,P ,, 119.9 Pxx,yy 1.240 Pxyqy 33.82 PyyYy 33.55 Pyy,,, 12.53 PylYl 10.51
Px,xx
5 7 9 Atomic Number Figure 1. Isotropic dipole polarizability (a) in au of AH, molecules vs. the atomic number of A. 3
This augmentation is specified in Table I. The composite basis sets, which we will designated as “ELP” (electrical properties) sets, are uniformly chosen and seem to be the minimum-sized basis sets for achieving reasonable accuracy for low order multipole polarizqbilities of small, first-row molecules. Permanent moments are reported at the S C F level and also at the well-correlated level of coupled cluster t h e ~ r y . ~ ~ The -~l permanent moments in general are more important in interactions than the polarizabilities, so relatively greater accuracy is called for. The same basis sets were used for the moments. ACCD42,43 was the particular coupled cluster approach and the moments were evaluated by expectation with the cluster expansion at single and double substitution^.^^
Results and Discussion Dipole polarizabilities through the third hyperpolarizability, 6, for the AH, molecules are given in Table 11. For LiH and from CH4 to HF, there is a regular decline in the isotropic dipole polarizability as illustrated in Figure 1. The uniform treatment of each molecule with respect to basis set quality assures that this trend is meaningful. A declining isotropic polarizability follows (33) Cizek, J. J. Chem. Phys. 1966, 45, 4256. Adv. Chem. Phys. 1969, 14, 35.
(34) Bartlett, R. J.; Purvis, G. D. In!. J. Quantum Chem. 1978, 14, 561, (35) Pople, J. A.; Krishnan, R.; Schlegel, H. B.; Binkley, J, S. Int. J . Quantum Chem. 1918, 14, 545. (36) Bartlett, R. J. Annu. Rev. Phys. Ckem. 1981, 32, 359. (37) Chiles, R. A.; Dykstra, C. E. J . Chem. Phys. 1981, 74, 4544. (38) Purvis, G. D.; Bartlett, R. J. J. Chem. Phys. 1982, 76, 1910. (39) Krasnicka, V.; Laurinc, V.; Biskupic, S.;Haring, M. Adu. Chem. Phys. 1982, 52, 181. (40) Bartlett, R. J.; Paldus, J.; Dykstra, C. E. In Advanced Theories and Computational Approaches to the Electronic Structure of Molecules, Dykstra, C . E., Ed.; Reidel: Dordrecht, 1984. (41) Jasien, P. G.; Dykstra, C. E. Int. J . Quant. Chem. Symp. 1983, 17, 189. (42) Chiles, R. A.; Dykstra, C. E. Chem. Phys. Left. 1981, 80, 69. Bachrach, S.M.; Chiles, R. A.; Dystra, C. E. J. Chem. Phys. 1981, 75, 2270. (43) Jankowski, K.; Paldus, J. Int. J. Quantum Chem. 1980, 18, 1243.
18.51 0.0 2.202 2.202 10.02 0.0 0.0 10.02 0.0 18.51 2.202 10.02 18.51
-1.191 0.068 -0.230 3.966 0.526 0.930 2.548 1.152 0.698
-6.321 -0.567 -4.630 26.43 0.717 11.34 9.017 2.599 3.209
0.0 0.0 0.0 18.78 0.628 7.873 7.748 2.738 2.505
0.0 0.0 0.0 61.42 -0.596 18.09 20.68 9.078 5.800
-5.532 0.114 -0.567 38.95 0.200 11.98 12.33 4.948 3.704
transcisHZCCHZ HNNH HNNH
NH3
H20
-0.471 0.0 -0.913 -0.913 0.0 0.0 -1.883 -1.515 0.0 1.883 -1.515 1.883
-1.370 0.0 -1.460 0.138 0.0 0.0 0.0 -2.085 0.0 0.0 -0.393 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 5.724 0.0 0.0 0.0 1.389 6.618 0.0 0.0 -0.605 0.0 0.240
14.56 0.0 2.785 2.785 5.806 -0.836 0.836 5.806 0.836 11.70 0.903 5.399 11.70
5.883 0.0 1.664 1.878 3.552 0.0 0.0 2.050 0.0 6.420 1.068 2.388 6.765
63.24 0.0 6.859 0.635 38.282 0.0 0.0 23.62 0.0 35.22 3.524 14.56 3 1.606
32.44 -0.364 3.688 1.313 19.19 6.092 -1.685 13.46 -0.922 21.02 2.218 6.851 13.368
32.92 0.0 4.874 1.197 22.23 0.0 0.0 12.46 0.0 21.35 2.630 7.034 12.68
the filling of the n = 2 valence shell of the first-row atom until polarization has to be accomplished with n = 3 orbitals. Dipole polarizabilities for the molecules containing two first-row atoms are given in Table 111. The values of these molecules tend to follow those of the AH, molecules in the following sense. The d value of a given ABH, molecule is well estimated (to within about 15%) by the sum of the t? values for the A and B simple
1752 The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 TABLE VI: Hyperpolarizabilities (in au) of the Linear Molecules LiH HF CO Nz HCCH HCN -622.4 -420.4 -351.5 -136.5 -1316 -364.8 -475.8 2972 604.1 450.8 -540.3 5.805 -273.1 88 1.7 923.3 335.2 294.0 -12120 - 1608 -1946 -168.3 246.9 39.32 -1779 -522.1 -628.4 -5954 -806.2 -1 287
-31.63 -6.827 -9.313 -6.755 -28.94 -10.56 -9.191 22.79 0.230 2.561 -0.255 0.194 -0.225 2.955 1.648 0.653 0.497 -57.63 -5.124 -8.230 -4.983 -2.679 -1.152 -7.912 -3.171 -2.370 -45.70 -14.184 -7.716
-138.3 -25.40 -42.34 -21.69 -89.17 -25.82 -3 1.68 217.6 24.59 39.02 9.422 2.826 3.298 46.19 49.77 10.88 19.45 -788.0 -88.85 -134.1 -15.03 -10.43 -2.301 -1 18.3 -30.43 -43.95 -224.6 -48.68 -43.97
-106.3 -21.03 -35.12 -14.16 -67.16 -22.40 -22.38 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -425.1 -52.00 -75.29 -13.60 -9.172 -2.214 -62.22 -17.61 -22.30 -169.5 -46.30 -30.79
-332.9 -86.64 -137.8 -19.12 -324.2 -1 19.6 -102.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1329.0 -1 19.7 -263.6 -41.58 -23.28 -9.144 -275.1 -102.0 -86.58 -1027.0 -326.0 -175.1
-201.1 -42.67 -67.81 -16.46 -143.4 -49.60 -46.90 60.32 0.649 5,200 -2.738 -0.079 -1.330 2.698 4.104 2.81 1 0.646 -833.6 -82.99 -141.8 -34.1 1 -20.12 -6.996 -126.3 -42.17 -42.01 -387.8 -1 13.9 -68.47
hydrides, diminished by 6 au if there is a double bond between A and B, or diminished by 12 au if there is a triple bond: a(ABH,,) &(AH,) + &(AH,) - 6.0(NAB - 1) (2) where N A B is the A-B bond order. Thus, for HCN, one has &(est) = &(CH,) a(NH,) - 12 = 16.8 compared to the calculated value of 16.6. Table IV compares the estimated a’s with those computed.44 The accuracy of this simple estimator suggests that the isotropic polarizability is significantly determined by the heavy atoms, not the hydrogens, and that multiple bonds are, as expected, less polarizable. The anisotropy of the dipole polarizability, of course, has very much to do with the hydrogens. The first, second, and third dipole hyperpolarizabilities, /3, 7, and S in Table 11, are much more sizable for LiH than for the other AH,, molecules and this is associated with the ionic character of LiH. The principal /3 values for CH4, NH,, H 2 0 , and H F are all of comparable size. However, there is a definite diminishment in y and 6 values going from NH, to H 2 0 to HF. Table V lists the dipole-quadrupole and the quadrupolequadrupole polarizabilities of the molecules studied. Table VI lists the hyperpolarizabilities of the linear molecules as this is a compact list. Corresponding properties for the other molecules have been obtained as well. The axial (Cartesian) quadrupole polarizabilities, P,,,,,, show a decrease with atomic number in the AH, series which is almost linear. The same pattern is seen for C,,,,, the traceless-quadrupole polarizability.60 The P,,,,, values of A2H, molecules do not show any simple relation to corresponding values of AH,, molecules. The dipole polarizabilities of CHI, NH3, H20, HF, and C O were calculated with large basis sets by Werner and Meyer.2’ They also assessed basis sets effects and suggested means for selecting bases. Their SCF-level values are quite close to the values obtained here. For example, for water, they report & = 4.90 while our value is 4.75. Some small geometry differences enter into the comparison, but generally the agreement is good. Agreement with experimental dipole polarizabilities, for instance, for acetylene45.46and for n i t r ~ g e n , ~is’ better than 5%. Dynamic dipole
+
(44) A referee has suggested that improved agreement from eq 1 might result from a more quantitative or sophisticated measure of “bond order”. That remains to be explored. (45) Bogaards, M. P.; Buckingham, A. D.; Pievens, R. K.; White, A. H. J . Chem. Soc., Faraday Trans 1 1978, 74, 3008
Liu and Dykstra polarizabilities, along with w = 0 polarizabilities, have been obtained for CH,, NH,, HzO, and H F at the MC-SCF level by R e i n ~ c h . These ~ ~ values show correlation effects on the order of 10%of the S C F values. Roos and Sadlej have carried out extraordinarily complete calculations on the permanent moments of LiH29and the polarizabilitie~.~~ Our ELP basis values for the dipole polarizability tensor of LiH are all within 10% of their very large basis S C F results. The ACCD permanent dipole and quadrupole moments agree with their very large basis CAS-SCF (correlated) results to about 1%. Bishop and Maroulis have studied dipole and quadrupole poIarizabiIities and hyperpolarizabilities of HF with very large basis sets at the SCF Agreement with our economical ELP basis and their large basis results is within about 10% for a, /3, and y, and almost as good for the quadrupole polarizabilities. Sekino and Barlett51 have reported basis set studies on the a , /3, 7, and 6 values for HF and have given numerical Hartree-Fock values at equilibrium for axial tensor elements. They report axJ= 5.75, while our ELP basis value is 5.61; p,,,,, = 8.5, compared to the ELP basis results of 8.27; yxJ,xs= 270, compared to 263. Sekino and Bartlett carried out an important analysis to show the corrections to an equilibrium /3 that are implicit in the experimental determination. Dynamical effects, for one thing, are potentially quite important. Other ab initio calculations of polarizabilities of certain of the molecules studied here include those of Amos et al. on NH3,52 finite-field dipole and quadrupole polarizability calculations on C O by Diercksen and Sadlej,53and a and /3 determinations for H 2 0 by Bartlett and pur vi^.^^ The dipole polarizability and permanent moments of N2 have been nicely calculated by Morrison and Hay,23Amos,5Sand Mulden et al.56 The isoelectronic series N2, CO, and H C N dipole polarizabilities have been calculated by Gready et aL5’ Multipole polarizabilities of CH, have been reported by Diercksen and Sadlej,s8 and the dipole polarizabilities of the AH,, series have been calculated by Sadlej using efficient electric-field variant bases.59 Agreement of the sort
-
(46) Alms, G. N.; Burnham, A. K.; Flygare, W. H. J . Chem. Phys. 1975,
63, 3321. (47) Bridge, N. J.; Buckingham, A. D. Proc. R . Soc. London, Ser. A 1966,
295, 334. (48) Reinsch, E.-A. J . Chem. Phys. 1985, 83, 5784. (49) Roos, B. 0.; Sadlej, A. J. J . Chem. Phys. 1982, 76, 5444. (50) With our convention for P,,,,,, the relation to the (traceless) tensor dement CX,,, is cx,,x,= 2pxx,xx- 2PX,,, - 2prx,,, + i/2pyi,JY+ ‘/2p2z,2z+ PYY.ZZ.
(51) Sekino, H.; Bartlett, R. J. J . Chem. Phys. 1986, 84, 2726. (52) Amos, R. D.; Handy, N. C.; Knowles, P. J.; Rice, J. E.; Stone, A. J. J . Phys. Chem. 1985,89, 2186. (53) Diercksen, G. H. F.; Sadlej, A. J. Chem. Phys. 1985, 96, 43. (54) Purvis, 111, G. D.; Bartlett, R. J. Phys. Reu. 1981, 23, 1594. (55) Amos, R. D. Mol. Phys. 1980, 39, 1. (56) Mulder, F.; Van Dijk, G.; Van der Avoird, A. Mol. Phys. 1980, 39, 407. (57) Gready, J. E.; Bacskay, G. B.; Hush, N. S. Chem. Phys. 1978, 31, 461. ( 5 8 ) Diercksen, G. H. F.; Sadlej, A. J. Chem. Phys. Letr. 1985, 114, 187. (59) Sadlej, A. J. Mol. Phys. 1977, 34, 731. (60) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure IV; Van Nostrand-Reinhold: New York, 1979. (61) Meyer, W. J. Chem. Phys. 1973, 58, 1017. (62) Hellwege. K. H., Ed. Landolt-Bornstein, Vol. 7, New Series; Springer-Verlag: West Berlin, 1976. (63) Herzberg, G. Molecular Specrra and Molecular Strucrure 111; Van Nostrand-Reinhold: New York, 1966. (64) Herzberg, G. Molecular Spectra and Molecular Srrucrure I; Van Nostrand-Reinhold: New York, 1950. (65) Laurie, V. W.; Herschbach, D. R. J . Chem. Phys. 1962, 37, 1687. (66) Carlotti, M.; Johns, J. W. C.; Trombetti, A. Can. J . Phys. 1974, 52, 340.
Polarizabilities and Hyperpolarizabilities of AH,, and A2Hn TABLE VII: Permanent Moments" of AH, and A2H, Molecules
molecule LiH
HF
co
N2 HCCH HCN
cis-HNNH
moment rux
SCF
ACCD
2.366 6.015 D -7.329 -3.898 28.976 7.444 -6.267 1.006 -0.642 -1.632 D -6.739 -4.506 -0.048 -0.870 1.371 -0.792 -2.013 D -4.464 -3.131 -5.678 -0.236 -1.649 0.21 1 0.767 1.948 D -2.468 -4.247 -2.072 0.169 -0.091 -0.232 D -9.141 -7.592 8.798 1.459 -8.666 -7.780 -4.999 -10.424 -1.300 -3.303 D -6.822 -8.909 -9.050 0.332 -9.071 -9.218 -1 2.01 3 -9.624 -7.758 -9.353 2.771 1.204 3.059 D -9.180 -8.09 1 -9.338 4.250 -0.391 -1.298
2.308 5.868 D -7.220 -4.028 28.47 7.472 -6.309 0.989 -0.607 -1.544 D -6.956 -4.730 0.159 -0.788 1.310 -0.743 -1.890 D -4.621 -3.293 -5.837 -0.206 -1.654 0.210 0.7 15 1.818 D -2.639 -4.389 -1.795 0.221 0.049 0.126 D -9.161 -7.65 1 8.772 1.664 -8.780 -7.695 -5.325 -10.250 -1.195 -3.038 D -7.055 -8.804 -8.469 0.453 -9.246 -9.282 -1 1.840 -9.797 -7.918 -9.280 2.696 1.153 2.930 D -9.375 -8.258 -9.261 3.950 -0.838 -1.286
"Any values not listed are related by symmetry. Unless otherwise indicated, values are in au. Ethylene permanent moments were obtained with a basis smaller than ELP on carbon, the augmented functions being only one p set (a = 0.02) and two d sets (a = 0.9, 0.1). already discussed (e.g., 10% or so) between ELP basis S C F values reported here and any of the large basis SCF values is seen from these studies. It appears that the ELP bases are an economical set for moderately accurate determinations of low-order multipole electrical properties. Permanent moments are listed in Table VII. It is interesting that the correlation effects on the moments are generally small. Typically, correlation changes dipole moments and second moment
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 1753 TABLE VIII: Coordinates of the Atoms Relative to the Centers of Evaluation of the Electrical Properties (COM)
moleculeQ
atomic center
atomic coordinates, A X
V
0.200 376 8 -1.394 923 2 0.0 0.629 677 7 0.067 661 0 -0.3133669 -0.313 3669 0.065 569 2 -0.520 313 0 0.046 184 5 -0.870615 5 -0.638 998 6 0.479 401 4 -0.547 0.547 -0.601 0.601 -1.661 1.661 0.595 077 4 -0.558 022 7 -1.623 922 7 -0.669 500 0 0.669 500 0 -1.2299387 0.626 -0.626 0.923 983 4 -0.923 983 4 0.6255 -0.6255 1.0186166 -1.0186166
0.0 0.0 0.0 -0.629 677 7 0.0 0.937 5296 -0.468 7648 0.0 0.756 950 3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.9290498 0.0 0.0 0.983 864 8 -0.983 864 8 0.0 0.0 0.9585173 0.958 517 3
z
~
LiH CH4 NH3 H2O HF
co N2 HCCH
HCN
C2H4 trans-HNNH
cis-HNNH
0.0 0.0 0.0 -0.629 677 7 0.0 0.0 -0.811 9244 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
"The equilibrium bond lengths and angles used are as follows: LiH, R = 1.5957 Ai6' CH,, R = 1.0906 A;61NHS, R = 1.012 A, L H N H = 106.7°;62H20, R = 0.9572 A, LHOH= 104.52°;60HF, R = 0.9168 A;60 CO, R = 1.1 184 A;6oN2, R = 1.094 A;64 HCCH, R(CC) = 1.202 A, R(CH) = 1.060 A;63H C N , R ( H C ) = 1.0659 A, R ( C N ) = 1.1531 C2H4,R(CC) = 1.330 A, R ( C H ) = 1.076 A, LHCH = 116.6°;62transN2H2,R ( N N ) = 1.252 A, R ( N H ) = 1.028 A, L N N H = 106.85°;66 cisN2H2, R ( N N ) = 1.251 A, R ( N H ) = 1.036 A, L N N H = 112.3°.67 Other hydrogens are positioned equivalently. tensor elements by only a few percent. In many cases the third moment tensor elements show the same insensitivity to correlation. For small molecules, the higher moments sample the charge distribution with greater weighting in the outer regions of the molecule. This is where correlation affects the electron density the least, and so sensitivity to correlation effects of only a few percent is not unlikely. However, use of smaller bases, ones that would be poorer in the outer regions, could exaggerate the correlation effect because of basis set deficiency. This is probably why it is a common notion that correlation plays a more sizable role in determining electrical moments than found here.
Note Added in Proof. Two important s t ~ d i e s ~of* electrical ,~~ properties appeared after completion of our manuscript. Jameson and Fowler reported equilibrium SCF-level dipole and quadrupole polarizabilities of N2,HCN, HCCH, and diacetylene.68 Their values for N 2 and HCCH were very similar to the values included here from earlier work,14 and their H C N values are similar to values given here. They pointed out a sign inconsistency in ref 14, and that has been corrected in the tables herein. For polarization functions on the first-row atoms they used two dfunctions and one f-function and they explicitly assessed the role of the f-functions. Cernusak, Diercksen, and Sadlej69calculated properties of N 2 at S C F and correlated levels with a like polar(68) Jameson, C. J.; Fowler, P. W . J . Chem. Phys. 1986, 85, 3432. (69) Cernusak, I.; Diercksen, G. H. F.; Sadlej, A. J . Chem. Phys. 1986,
(67) Parsons, C. A,; Dykstra, C. E. J . Chem. Phys. 1979, 7 1 , 3025.
108, 45.
J. Phys. Chem. 1987, 91, 1754-1760
1754
ization set, but contracted from more primitives. Generally, their SCF-level results for quadrupole-related polarizabilities (e.g., B, C, D) are smaller in magnitude than the values given here or given by Jameson and Fowler. Also, Cernusak, Diercksen, and Sadlej incorrectly compared their results to those of DykstraI4 by failing to appreciate that their values were in traceless form while Dykstra's were not. This also led them to assert improperly that there were errors in numerical factors and that symmetry rules were not satisfied in that work.I4 In fact, all symmetry properties of the Cartesian tensors reported14 were satisfied, and when the Cartesian tensors are converted to traceless form and then properly compared with the results of Cernusak, Diercksen, and Sadlej, the differences that remain are not what was i n d i ~ a t e d .Jameson ~~ and Fowler68presented the simple transformation expression from traced to traceless form for the B tensor and presented values both ways. Applequist has discussed relationships and transformations of this sort in detail.'O An example comparison for the three studies is the traceless tensor element B , , , , for N, where the values reported herein yield -170.6 au. Jameson and Fowler obtained -174 au and Cernusak et al. obtained -141.8 au at the S C F level. (70)Applequist, J. Chem. Phys. 1984, 85, 279.
Acknowledgment. The support of the National Science Foundation (Grant C H E 84- 19496) is gratefully acknowledged. Appendix
The electrical properties of molecules are sensitive to the structural parameters. For instance, certain hyperpolarizability tensor elements of LiH change quickly even in the vicinity of the equilibrium structure.6 For all molecules, an equilibrium structure has been used. These structures are specified in Table VIII. The Cartesian coordinates of the atomic centers are listed because certain of the properties are not invariant to the origin choice for evaluating the properties. The origins that have been chosen are center-of-mass locations for the most abundant isotopic forms. The polarizabilities and hyperpolarizabilities are equilibrium electronic properties. Laboratory measurement of a dipole polarizability would determine the second derivative of the electronic-vibrational state with respect to an applied field and this includes terms other than those of the equilibrium electronic re~ponse.~~*~~~' Registry No. LiH, 7580-67-8; CH4, 74-82-8;NH3, 7664-41-7; HzO, 7732-18-5;H F , 7664-39-3;CO,630-08-0; N2, 7727-37-9;HCCH, 7486-2;H C N , 74-90-8;H2CCH2, 74-85-1;Zrans-HNNH, 15626-43-4; cis-HNNH, 15626-42-3.
Optical and Magnetic Properties of the Lowest Triplet State of Pyrido(2,3-b)pyrazine. An Example of 'n7r* Azanaphthalene Seigo Yamauchi* and Noboru Hirota Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan (Received: July 25, 1986; In Final Form: October 25, 1986)
The lowest excited triplet (TI) states of pyrido(2,3-b)pyrazine (PP; 1,4,5-triazanaphthaIene)in three kinds of environments have been studied by using various spectroscopic and magnetic resonance techniques. We obtained phosphorescenceemission and excitation spectra, triplet lifetimes, zero field splittings (zfs), and triplet sublevel properties. From the analysis of the results we conclude that T I is 3n1r* in character in a single crystal of durene and in ethanol, but it is 3 7 r ~ *in trifluoroethanol. Observation of the 3n1r* phosphorescence was made for the first time in azanaphthalenes, which provides direct information about 3 n ~ azanaphthalene. * The 0-0 band and vibronic bands involving a' vibrations are dominant in the phosphorescence spectrum. Large zfs and negative D (= -3/2X; -3.1 GHz) were obtained. As for the sublevel properties T, is the most active and T, is inactive in both radiative and nonradiative processes, whereas Ty is moderately active in the T I Sononradiative * were discussed in comparison with those of 31r7r* PP and azanaphthalenes and 3n7r* decay process. These 3 n ~ properties azabenzenes. Possible mechanisms to explain the radiative and nonradiative properties are given.
--
1. Introduction The lowest excited triplet (TI) states of azanaphthalenes have been investigated extensively by using their relatively strong phosphorescence and EPR signals. It is known that all the phosphorescent TI states of azanaphthalenes are 31r7r* in character,]-I0 which include those of quinoline( 1-), isoquinoline(2-), (1) Schmidt, J.; Veeman, W. S.; van der Waals, J. H. Chem. Phys. Lett. 1969, 4, 341.
(2) Clarke, R.H.; Hayes, J. M. J . Chem. Phys. 1973,59, 3113. 1972.57, 569. (3) (a) Vincent, J. S.; Maki, A. K. J. C h e p . Phys. 1963,39,3085. 1965, 42, 865. (b) Vincent, J. S . Chem. Phys. Lett. 1971, 8, 37. (4)Schmidt, J.; Antheunis, D. A.; van der Waals, J. H. Mol. Phys. 1971, 22, 1.
( 5 ) Yamauchi, S.; Azumi, T. J . Chem. Phys. 1977, 67, 7. (6)Suga, K.; Kinoshita, M. Bull. Chem. SOC.Jpn. 1982, 55, 1695. Tinti, D. S.;Vincent, J. S. Chem. Phys. Lett. 1971, (7)Nishimura, A. M.; 12, 36. ( 8 ) Dennis, D. W.; Tinti, D. S . J . Chem. Phys. 1975, 62, 2015.
quinazoline( 1,3-), quinoxaline( 1,4-), phthalazine(2,3-), 1,5-, 1,6-, and 1&diazanaphthalenes, and pteridine (1,4,5,7-tetraazanaphthalene). The phosphorescence properties of these compounds are very similar. Recently, we have investigated the nonphosphorescent T1 state of phthalazine by a time-resolved EPR (TREPR) technique" and found that T I possesses a heavily mixed character of 3n7r* and 3mr*. We explained the changes in the zero field splitting (zfs) and hyperfine splitting (hfs) in different surroundings in terms of vibronic mixing between 3n7r* and 37r1r* states. However, in azanaphthalenes no phosphorescence has been reported for a triplet state with a dominant 3n7r* character. Therefore, very little is known about the spectroscopic properties of 3n7r* azanaphthalenes. Since vibronic and/or spin-orbit couplings between 3 ~ 7 r *and 3n7r* states are often invoked in the (9) Bramley, R.; McCool, B. J. Mol. Phys. 1975, 29, 649. ( I O ) Brian, J.; Markey, B. R.;Bramley, R. Mol. Phys. 1984, 51, 935. (1 1) Terazima, M.; Yamauchi, S.; Hirota, N. J . Chem. Phys. 1985, 83,
3234.
0022-365418712091- 1754$01.5010 0 1987 American Chemical Society
