J. Phys. Chem. 1996, 100, 6359-6361
6359
Quantum Chemical Calculations of Infrared and Raman Intensities for Diatomics on the Basis of the Virial Theorem V. V. Rossikhin,* E. O. Voronkov, and V. V. Kuz’menko Department of Physics, State Technical UniVersity, 2 Acad. Lazaryan St., 320010 DnepropetroVsk-10, Ukraine
Yu. A. Kruglyak Department of Molecular Electronics, Odessa State UniVersity, 2 DVoryanskaya St., 270100 Odessa, Ukraine ReceiVed: September 20, 1995; In Final Form: February 21, 1996X
An ab initio method of calculations of the analytic dipole moment and polarizability geometric derivatives for diatomics by means of differentiation of expression for the quantum mechanical virial theorem in the presence of an external field is suggested. Within the framework of the Hartree-Fock perturbation theory, a dependence of both the density matrix and basis functions on the external field strength is taken into account. The results of our test calculations for some diatomics are in satisfactory agreement with the prior theoretical calculations as well as experimental data.
Introduction The theoretical study of molecular spectra and electric properties is becoming an invaluable tool in the identification of chemical compounds. So, geometrical derivatives of molecular dipole moments and polarizabilities are of great interest due to their relation to infrared absorption and Raman scattering intensities. Several theoretical approaches are widely used nowadays for calculation of such derivatives, e.g., derivative Hartree-Fock theory,1,2 perturbation theory,3 finite-field difference techniques,4,5 and the method of nuclear electric shielding tensor.6,7 These methods have, in our opinion, one of two essential deficiencies. First, they take place because of difficulties in procedures of numerical differentiation of the molecular potential energy surface in the absence or presence of an external field discussed in detail by us.8 Second, analytical approaches require cumbersome differentiation of two-electron, many-center integrals. With account of this precondition, it is of interest to discuss possibilities arising owing to the use of the virial-relationships-based method for calculation of derivatives of some molecular electric and magnetic characteristics with respect to parameters dependent on the nucleus coordinates. In this paper a method of ab initio determination of such a kind of derivative on the basis of the quantum mechanical virial theorem for the molecular electronic energy in external uniform field in the single-determinant approximation of the HartreeFock-Roothaan theory is suggested and applied for diatomic molecules.
wave function if a scale factor has been determined through the variation principle. Formulas for the dipole moment D and polarizability R derivatives with respect to R can be obtained by the direct differentiation of (1) with respect to F:
∂D 1 ) (D + 2〈ψ(0),Tˆ 0ψ(1)〉) ∂R R
(2)
∂R 1 ) (3R + 2〈ψ(1),Tˆ 0ψ(1)〉 + 4〈ψ(0),Tˆ 0ψ(2)〉) ∂R R
(3)
For molecules with the closed shells the approximate wave function ψ(0) can be expressed in the form of a single determinant: (0) (0) ψ(0) ) (N!)-1/2 det |ψ(0) 1 ,ψ2 ,..., ψN |
constructed on the Hartree-Fock spin orbitals ψ(0) i . Then for the first- and second-order correction wave functions ψ(1) and ψ(2), respectively, (F is treated as perturbation) we have N
(0) (1) (0) ψ(1) ) (N!)-1/2∑det |ψ(0) 1 ,ψ2 ,...,ψi ,...,ψN |
∂E ∂E + E - 2F + 〈ψ,Tˆ 0ψ〉 ) 0 ∂R ∂F
R
N
(0) (2) (0) ψ(2) ) (N!)-1/2(∑det |ψ(0) 1 ,ψ2 ,...,ψi ,...,ψN | + i)1 N
(0) (1) (1) (0) det |ψ(0) ∑ 1 ,ψ2 ,...,ψi ,...,ψj ,...,ψN |) i>j)1
X
Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-6359$12.00/0
(6)
where
ψi(n) )
(1)
where R is the internuclear distance, Tˆ 0 is the kinetic energy operator, F is the electric field strength, E is the molecular electronic energy, and ψ is the eigenfunction of the total Hamiltonian. Formula 1 holds true also for the approximate
(5)
i)1
Theoretical Approach In the adiabatical approximation the virial theorem for a diatomic in the presence of an external electric field has the following form:9
(4)
( )
n 1 ∂ ψi n! ∂Fn
(7) F)0
With account of (4)-(6) after summing up the spin variables, for the spin-independent Hamiltonian formulas 2 and 3 can be presented in the form
2 ) (-∑〈φi(0),dφi(0)〉 + 2∑〈φi(0),Tφi(1)〉) ∂R R i i
∂D
© 1996 American Chemical Society
(8)
6360 J. Phys. Chem., Vol. 100, No. 15, 1996
Rossikhin et al.
A ) ∑B Xτ‚∇ B τ + ∑b x ν‚∇ Bν
4 ) [-3∑〈φi(0),dφi(1)〉 + ∑(〈φi(1),Tφi(1)〉 + ∂R R i i
∂R
τ
〈φi(0),Tφi(0)〉〈φi(1),φi(1)〉) + 2∑(〈φi(0),Tφi(2)〉 + i
〈φi(0),Tφi(0)〉〈φi(2),φi(0)〉)] (9) (0) (0) (1) In (8) and (9) the terms like ∑i,j〈φ(0) i ,Tφj 〉〈φj ,φi 〉 are omitted because the functions satisfy the following conditions:
B Xτ and b xν are the nucleus and electron position vectors, respectively. Summing up j in formula 16 can be transformed into the form
∑j ′〈φ(0)k ,dφj(0)〉〈φj(0),Aφi(0)〉 ) (0) (0) (0) (0) (0) 〈φ(0) k ,dAφi 〉 - 〈φk ,dφi 〉〈φi ,Aφi 〉 (18)
n
〈φi(m),φj(n-m)〉 ) δijδ0n ∑ m)0
(10)
Finally, the first term of (14) with a substitution of (16) and (18) is
From (10) it also follows that
〈φi(2),φi(0)〉
(I) ) 〈φi(1),dAφi(0)〉 - 〈φi(0),Aφi(0)〉〈φi(1),dφi(0)〉
1 ) - 〈φi(1),φi(1)〉 2
(11)
∂R
)
4
∑i
R
〈φi(1)| (-3〈φi(0),dφi(1)〉
+
〈φi(1),Tφi(1)〉
+2
〈φi(0),Tφi(2)〉) (12)
j k (0) 〈φk ,dφi(0)〉|φj(0)〉
∂R
j
〈φj(0),dφi(0)〉|φj(0)〉
-
∑j
2
′(i(0)
-
j(0))-2|〈φj(0),dφi(0)〉|2
(13)
This form corresponds to one of variants of the so-called uncoupled Hartree-Fock perturbation theory.11,12 Then for the matrix element we have got (0) (0) (0) (0) 〈φj(0),dφ(0) k 〉〈φk ,dφi 〉〈φi ,Tφj 〉
〈φi(0),Tφi(2)〉 ) ∑′∑′ j
-
(0) (0) (i(0) - (0) k )(i - j )
k
〈φj(0),dφi(0)〉〈φi(0),Tφj(0)〉
∑j ′
〈φi(0),dφi(0)〉
-
(i(0) - j(0))2 〈φi(0),Tφi(0)〉
|〈φj(0),dφi(0)〉|2
2
(i(0) - j(0))2
∑j ′
(14)
The first term in (14), using the relationship
〈φj(0),Tφi(0)〉 ) (i(0) - j(0))〈φj(0),Aφi(0)〉
(20)
(II) ) 〈φi(0),dφi(0)〉〈φi(1),Aφi(0)〉
(21)
(III) ) 1/2〈φi(0),Tφi(0)〉〈φi(1),φi(1)〉
(22)
By means of substituting (19), (21), and (22) in (12), we derive the expression as
- ∑′(i(0) - j(0))-2〈φi(0),dφi(0)〉 ×
|φi(0)〉
(0) 〈φ(0) k ,dφi 〉 ) ∑′ 〈φ(0) k | k (0) - (0) i k
The second and third terms in (14) can be represented by analogy with the first one, i.e.
This formula is more difficult to use as compared to (8) because of the presence of φ(2) in the last term. The procedure i suggested here enables us to overcome this problem. Let us 10 write φ(2) i as in the standard perturbation theory: (0) -1 (0) (0) -1 (0) |φi(2)〉 ) ∑′∑′(i(0) - (0) k ) (i - j ) 〈φj ,dφk 〉 ×
(19)
where it has been accounted that
Taking it into account, (9) can be expressed in such a form:
∂R
(17)
ν
(15)
obtained by us13 from the nondiagonal virial theorem for the single-determinant wave function constructed with the HartreeFock spin orbitals, can be expressed as
∂R
)
4
[ (
∑i -3〈φi(0),dφi(1)〉 + 〈φi(1),Tφi(1)〉 +
R
2 〈φi(1),dAφi(0)〉 - 〈φi(0),Aφi(0)〉〈φi(1),dφi(0)〉 〈φi(0),dφi(0)〉〈φi(1),Aφi(0)〉
〈φi(0),Tφi(0)〉 〈φi(1),φi(1)〉 2
)]
(23)
Thus, the numerical realization of (23) as well as (9) requires (1) the knowledge of the unperturbed φ(0) i and first-order φi wave functions only. Moreover, these formulas do not require the differentiation procedure, since the geometric derivatives of dipole moments and polarizabilities are expressed through the corresponding properties and the matrix elements of the oneelectron operator of kinetic energy, which can be calculated much easier than the potential energy operator matrix elements and their derivatives. As compared to traditional approaches, our method enables decreasing by one the order of differentiation of the total energy for calculation of the second- and third-order properties. In the algebraic variant of theory14 orbitals φi can be approximated as expansion over finite-dimension AO basis functions χp:
φi ) ∑Cipχp
(24)
p
(0) 〈φ(0) k ,dφi 〉
∑k ′
i(0)
-
(0) k
∑j ′〈φ(0)k ,dφj(0)〉〈φj(0),Aφi(0)〉
where A is an operator of the form
(16)
where orbital coefficients Cip and χp depend in general on the perturbation parameter λ. Therefore, for φ(λ) i we have (λ) φi(λ) ) ∑(Cip χp + Cipχ(λ) p ) p
(25)
Infrared Intensities for Diatomics
J. Phys. Chem., Vol. 100, No. 15, 1996 6361
TABLE 1: Derivatives of Dipole Moments and Polarizabilities for Diatomics molecule
method
H2+
our calc ref 17 ref 18 our calc ref 19 exp our calc SCF CI exp our calc SCF22 CI23 exp our calc exp19
LiH HF21
CO
N2
∂D/∂R (D/Å)
-2.50 -18.7720 (2 ( 0.3 1.69 1.87 1.54 1.52 5.01 4.93 4.99
∂Rzz/∂R (Å2)
∂Rxx/∂R (Å2)
∂Rav/∂R (Å2)
6.701 6.850 7.307 7.140 5.050
1.120 1.260 1.150 0.560 0.320
2.980 3.120 3.202 2.750 1.897
1.55 1.48 1.49
0.292 0.294 0.291
9.87 9.55 10.51
2.57 2.70 2.39
5.17
0.42
0.711 0.689 0.692 0.7-1.0 5.00 4.98 5.10 5.36-5.54 2.00 1.75
Appendix
Calculation of C(λ) ip is carried out on the base of expression obtained by us:13 m
(λ) Cip
)∑
m
Kij(λ)Cjp
j)1
-
∑
(λ) -1 Ciqµqr (S )rp
atomic molecules. It enables us to obtain the results with sufficient accuracy and considerable saving of computer time. Now we are developing a computer program for calculation of geometric derivatives of these properties.
In conclusion, we would like to note that relationships similar to (2), (3), etc., can be obtained for a case of external magnetic field. Actually, the virial theorem for the electronic energy of diatomic molecule in the presence of external uniform magnetic field with strength H can be written in the form
∂E + E + 〈Ψ,TΨ〉 + H〈Ψ,MΨ〉 - 3H2〈Ψ,WΨ〉 ) 0 ∂R (A.1)
R
where M ) (-e/2mc)L is the magnetic moment operator, L is r b]. the angular momentum operator, and W ) (e2/8mc2)[r2 - b‚r The direct differentiation of expression A.1 with respect to the magnetic field strength H leads to
∂M 2 (1) ) 〈Ψ ,TΨ(0)〉 ∂R R
(26)
q,r)1
where (λ) ) 〈χ(λ) Kij(λ) ) 〈φi(λ),φj〉, µqr q ,χr〉
Let us note that if λ is the external electric field strength, this corresponds to (7), whereas the coefficients Kij(F) can be calculated as is done in the standard perturbation theory. Correction functions χ(F) p are determined in ref 15 by means of perturbation theory with use of the Green’s function closed representation.16
(A.2)
∂κ 1 ) (κ - 2〈Ψ(0),WΨ(0)〉 + 2〈Ψ(1),T0Ψ(1)〉 + ∂R R 4〈Ψ(2),T0Ψ(0)〉) (A.3) where M is the molecular magnetic moment, κ is the magnetic susceptibility, and Ψ(0), Ψ(1), and Ψ(2) are the unperturbed, firstorder and second-order on H wave functions, respectively. Within the framework of approach suggested in accordance with the procedure described above, it is easy to derive the working formulas for the magnetic moment and susceptibility derivatives with respect to R for a diatomic molecule. References and Notes
Results and Discussion The outlined approach has been applied to the calculation of ∂D/∂R and ∂d/∂R for a number of diatomics in the 6-31G splitvalence basis sets. Results of test calculations are presented in Table 1 and compared (where possible) with experiment and results of other authors. First of all, the data demonstrate a sufficiently favorable quality. Let us note that the method suggested in the given approximation enables us to use in fact one and the same (in type and size) set of basis functions for both the unperturbed and perturbed molecule. In our opinion, it is caused by the correct analytical treatment of the dependence of the orbital coefficients as well as the basis functions on field strength. From the results presented in Table 1 it follows that the contribution of the correlation effects to values of at least firstorder derivatives of the dipole moments and polarizabilities with respect to corresponding coordinates (i.e., within the region of electrooptical harmonicity) is substantially small in comparison with the CI contributions for the dipole moments and polarizabilities, where they are in the range of 20-30% of total values. So, we may conclude that the first-order geometric derivatives calculation can be carried out within the framework of the SCF method but with the basis sets of proper size and quality. As for computer realization of our general formalism, at present we have the POLMAG program developed for calculation of polarizabilities and magnetic susceptibilities of poly-
(1) Malik, D. J.; Dykstra, D. E. J. Chem. Phys. 1985, 83, 6307. (2) Malik, D. J.; Dykstra, D. E. J. Chem. Phys. 1987, 87, 2806. (3) Bishop, D. M. ReV. Mod. Phys. 1990, 62, 343. (4) Adamovicz, L.; Bartlett, R. J. J. Chem. Phys. 1986, 84, 4988. (5) Luis, J. M.; Marti, J.; Duran, M.; Andres, J. L. J. Chem. Phys. 1995, 102, 7573. (6) Lazzeretti, P.; Zanasi, R. J. Chem. Phys. 1985, 83, 1216. (7) Lazzeretti, P.; Zanasi, R.; Prosperi, T.; Lapiccirella, A. Chem. Phys. Lett. 1988, 150, 515. (8) Rossikhin, V. V.; Morozov, V. P. Force Constants and Electrooptical Parameters of Molecules; Energoatomizdat: Moscow, 1983. (9) Epstein, S. T. The Variation Method in Quantum Chemistry; Academic Press: New York, 1974. (10) Landau, L. D.; Lifshitz, E. M. Quantum Mechanics; GIFML: Moscow, 1963. (11) Dalgarno, A. Proc. R. Soc. London 1959, A251, 282. (12) Dalgarno, A.; McNamee, J. M. J. Chem. Phys. 1961, 35, 1517. (13) Rossikhin, V. V.; Bolotinas, A. B.; Zaslavskaya, L. I. Liet. Fiz. Rink 1972, 12, 753. (14) Roothaan, C. C. J. ReV. Mod. Phys. 1951, 23, 69. (15) Bolotinas, A. B.; Voronkov, E. O.; Zaslavskaya, L. I.; Rossikhin, V. V. Liet. Fiz. Rink 1990, 30, 677. (16) Hostler, L. J. Math. Phys. 1964, 5, 591. (17) Adamov, M. N.; Rebane, T. K.; Evarestov, R. A. Opt. Spektrosk. 1967, 22, 709. (18) McEachran, R. P.; Smith, S.; Cohen, M. Can. J. Chem. 1974, 52, 3463. (19) Slepukhin, A. Yu.; Kovner, M. A.; Guriev, K. I. Theor. Eksp. Khim. 1973, 9, 799. (20) Gerrat, J.; Mills, I. J. Chem. Phys. 1968, 49, 1730. (21) Amos, R. D. Mol. Phys. 1978, 35, 1765. (22) Kirby-Doken, K.; Liu, B. J. Chem. Phys. 1977, 66, 4309. (23) Amos, R. D. Chem. Phys. Lett. 1980, 70, 613.
JP952813T
