J. Phys. Chem. 1991,95,4255-4262
4255
Theory of Vibrational Circular Dichroism: Formalisms for Atomic Polar and Axial Tensors Using Noncanonlcai Orbitals Aage E. Hansen,* Department of Physical Chemistry, H . C. 0rsted Institute, Universitetsparken 5, DK-2100 Copenhagen 0, Denmark
Philip J. Stephens,* Department of Chemistry, University of Southern California. Los Angeles, California 90089
and Thomas D. human* Department of Chemistry, Southern Illinois University, Edwardsville, Illinois 62026 (Received: August 27, 1990)
Expressions for atomic polar and axial tensors and, thence, vibrational dipole and rotatory strengths are derived from the random phase approximation (RPA). These expressions permit the use of either canonical or noncanonical molecular orbitals and, in the case of atomic axial tensors, incorporate a flexible choice of magnetic gauge. In particular, they allow the use of localized molecular orbitals together with individual gauge origins for individual orbitals in calculating atomic axial tensors, and hence vibrational rotatory strengths and circular dichroism spectra, in a linear response formulation analogous to that recently employed in the calculation of nuclear magnetic shielding tensors. The approximate LMO model of Nafie and mworkers and the STA model of Dutler and Rauk are discussed in the context of the above derivations.
I. Introduction Vibrational absorption and circular dichroism intensities are governed by the atomic polar and axial tensors, papand MAa8, which determine, respectively, the electric and magnetic dipole transition moments.'-' These tensors can be efficiently calculated ab initio by using analytical derivative methods involving the solution of the coupled Hartree-Fock (CHF) perturbation equation^.^'^ Alternatively, the tensors expressed in terms of sums over states have been evaluated by using the random phase approximation (RPA).'6*wB Implementation of the analytical ~~~
~~
~
( I ) (a) Person. W.B. In Vibrational Inlensities in Infrared and Raman Spectmcopy Person, W. B.. Zerbi, G.,Eds.; Elscvier: Amsterdam, 1982. (b) Person, W. B.; Newton, J. H. J. Chem. Phys. 1974, 61, 1040. (2) Stephens, P. J. J . Phys. Chem. 1985,89,748. (3) Stephens, P. J. J . Phys. Chem. 1987, 91, 1712. (4) Stephens, P. J. Croat. Chem. Acta 1989, 62, 429. (5) Amos, R. D. Adv. Chem. Phys. 1987,67,99. (6) M a y , P. Adu. Chem. Phys. 1987,69, 241. (7) Amos, R. D. Chem. Phys. Leu. 1984,108, 185. (8) Yamaguchi, Y.; Frisch, M.; Gaw, J.; Schaefer, H. F.; Binkley, J. S. J. Chem. Phys. 1986,84,2262. (9) Amos, R. D.; Handy, N. C.; Jalkanen, K. J.; Stephens, P. J. Chem. Phys. Lett. 1987, 133. 21. (IO) Jalkanen, K . J.; Stephens, P. J.; Amos, R. D.; Handy, N. C. J. Am. Chem. Soc. 1987,109,7193. ( I I ) Jalkanen, K. J.; Stephens, P. J.; Amos, R. D.; Handy, N. C. Chem. Phys. Lett. 1987, 142, 153. (12) Jalkanen, K. J.; Stephens, P. J.; Amos, R. D.; Handy, N . C. J . Phys. Chem. 1988,92, 1781. (13) Jalkanen, K . J.; Stephens, P. J.; Amos, R. D.; Handy, N. C. J . Am. Chem. Soc. 1988,110,2012. (14) Kawiecki. R. W.; Devlin, F.; Stephens, P. J.; Amos, R. D.; Handy, N. C. Chem. Phys. Lert. 1988, 145,411. (IS) Amos. R. D.; Jalkanen, K. J.; Stephens, P. J. J . Phys. Chem. 1988, 92, 5571. (16) Stephens, P. J.; Jalkanen, K. J.; Amos, R. D.; Laueretti, P.; Zanasi, R. J. Phys. Chem. 1990, 94, 181 I . (17) Jalkanen, K. J.; Kawiecki, R. W.; Stephens, P. J.; Amos, R. D. J. Phys. Chem., in press. (18) Stephens, P. J.; Jalkanen, K. J.; Kawiecki, R. W. J . Am. Chem. Soc., in press. (19) Ama, R. D.; Handy, N. C.; Drake, A. F.; Palmieri, P. J. Chem. Phys. 1988.89,7281.
derivative CHF and RF'A methods for ab initio calculation of MAd tensors has made clear the critical importance of the choice of magnetic gauge to the accuracy of results obtained from finite basis sets, especially those of modest size. A gauge choice in which each nucleus carries its own gauge origin, termed the distributed origin (DO) gauge,' has been introduced by Stephens and shown tensors, to yield marked improvement in the accuracy of MAors relative to results obtained from a single gauge origin (the common origin (CO) gauge3).'2J6J7~27In the DO gauge, one has the additional advantage that calculated vibrational circular dichroism (VCD) intensities are independent of the choice of molecular coordinate origin.' CHF and RPA methods are also widely used in the calculation of other magnetic properties, such as magnetic susceptibility tensors xag and nuclear magnetic shielding tensors Qxd, which are similarly gauge-dependent. For these properties, two different types of noncentral gauge origin choice have been used to improve upon calculations using the CO gauge. The earliest of these is the GIAO method, in which each atomic orbital is assigned its own gauge origin, defined as the position of the nucleus carrying that ~ r b i t a l . ~More ~ , ~recently, the IGLO (individual gauge for localized orbital) method of Kutzelnigg and Schindler,"-" and the LORG (localized orbital/local origin) method of Hansen and B o ~ m a n , ' ~both ' ~ choosing a gauge in which each localized ~~
(20) (21) (22) (23) (24) (25) (26)
Lazzereii, P.; Zanasi, R.; Bursi, R. J . Chcm. Phys. 1985.83, 1218. Lazzeretti, P.; Zanasi, R. Phys. Reu. A 1986, 33, 3727. Lazzeretti, P.; Zanasi, R. J . Chem. Phys. 1986,84, 3916. Lazzeretti, P.; Zanasi, R. J. Chem. Phys. 1986,85, 5932. Lazzeretti, P.; Zanasi, R. J . Chem. Phys. 1987, 87, 472. Lazzeretti, P.; Zanasi, R.; Bursi, R. J . Chem. Phys. 1988,89, 987. Lazzcrctti, P.; Zanasi, R.; Prospcri, T.; Lapiccirella, R. Chem. Phys.
1988, 150, 515. (27) Jalkanen, K. J.; Stephens, P. J.; Lazzeretti, P.; Zanasi, R. J. Chem. Phys. 1989, 90,3204. (28) Jalkanen, K. J.; Stephens, P. J.; Lazzeretti, P.; Zanasi, R. J. Phys. Chem. 1989, 93,6583. (29) (a) Ditchfield, R. In MTP Internarional Reuiew of Science Physicof Chemistry Series I. Molecular Structure and Properties, Butterworthr: London, 1972, and references therein. (b) Ditchfield, R. Mol. Phys. 1974,
kit.
-57., 789. (30) Chesnut, D. B.; Foley, C. K. Chem. Phys. k i t . 1985, 118. 316. (31) Kutzelnigg, W. Isr. J. Chem. 1980. 19, 193. (32) Schindler, hi.; Kutzelnigg, W. J . Am. Chem. Soc. 1983, 105, 1360. (33) Schindler, M.; Kutzclnigg, W. J . Chem. Phys. 1982, 76, 1919. (34) Flcischer, U.; Schindler, M.Chem. Phys. 1988, 120, 103.
0022-365419112095-4255$02.50/0 0 1991 American Chemical Society
Hansen et al.
4256 The Journal of Physical Chemistry, Vol. 95, No. 11, 1991
(occupied) molecular orbital carries its own gauge origin, have been widely applied, especially for NMR shieldings (see ref 38 for a comparison of the latter two methods). Our primary goal here is to develop a LORG formalism for the calculation of VCD intensities. The development will start from the RPA formulation for atomic polar and axial tensors, allowing the use of canonical or noncanonical (e.g., localized) molecular orbitals, eventually arriving at expressions that can be evaluated without explicit reference to solution of the RPA equations. This route will let us establish connections both to the analytical derivative CHF approach of Amos et al? and to the RPA approach of Lazzeretti and co-worker~.~'The resulting LORG formalism also lets us place the localized molecular orbital (LMO) approach of Nafie and co-worker~'"*~'in the proper perspective. In addition, the localized orbital expressions for atomic polar and axial tensors and for vibrational intensities provide decompositions into "bond" contributions and open the possibility of new, more flexible gauge choices combining aspects of both the DO and the LORG gauges.
with EAu) [a(*~(R)IccC,igl*~(R)) /~X,IR,, 2( ( ~ ~ O ( R ) / ~ X ~ ) ~ C C ' C I J ~) *(8) G(RO)
and NAu@= eZA6&
(9)
The magnetic dipole transition moment is (OICrma,+4)'Il
)i
= -(2h3~i)'/2CS,,jMAup(r) A#
= - ( 2 h 3 4 ' / 2 C ~ A ~ ~=( -r()2 h 3 0 i ) l / 2 ~ i a ( r()i o ) A
indicating explicitly that the magnetic dipole moment operator and the atomic axial tensor MA4depend upon the choice of the arbitrary gauge origin l'. The atomic axial tensor MA4is again a sum of an electronic and a nuclear part
+ JAup(r)
MAq9(r) = fA#)
(11)
11. General Intensity relation^*-^*.'^
where the electronic part is
Within the harmonic approximation, the absorption intensity in unpolarized light for a fundamental vibrational transition in the ith normal mode is governed by the dipole strength
IAu#) ( [a*G(R) /aX~al&l[a*G(bHg> /aH,91~~=0) ( 12)
Di(*l)
I(oI~e111 )i12
(1)
Here +G(&,HB) is the electronic ground-state wave function at the equilibrium geometry & in the presence of the uniform magnetic field perturbation H' = (e/2mc)H&[(ri - I")
while the circular dichroism of the transition is governed by the rotatory42 strength Ri(O-*l) = Im [(Ol*e111
)t(1l~maglO)il
(2)
The electric dipole transition moment is required for both of these intensities and can be expressed as A,a
( h / 2 W i ) ' ~ 2 C P j=q (h/2@,)'/2Pi@ (3) A
where a and fl are Cartesian indices, X is a nuclear index, and SAa,/defines the transformation between Cartesian nuclear displacements and normal coordinates:
zsx&!i i
(4)
where X , = Rxa- Pk, and RA and Rox denote the displaced and equilibrium positions of nucleus A. The atomic polar tensor PA.@=
[a(*G(R)lr(el,@I*G(R))
/aXAulRo
(5)
expressed in terms of the electronic ground-state wave function (a function of nuclear coordinates R,Ro at equilibrium) and of the electric dipole operator pclJ
+
peel,@ pneI,@I -exrip i
+ ezZARA,9 A
(6)
PAup= EAu@ NAap
JAu@(r) = (iezA/4hc)C%p7(pA7- r7)
(13)
(14)
where cup7. is the antisymmetric unit tensor. Expanding +G(R) and + ~ ( k H b to ) first order in X, and HB, respectively, we can write eqs 8 and 12 in the conventional sumover-states versions from second-order perturbation theory: EA&) = 2e (*GlaHel/aXAul*q)&(vq - vG)-'(*qlTri@l*G)& q+G
(15)
fAUs(r) = ( e / 2 m c ) C (*GlaHeI/aXd*q)&' q*G
(vq - wd-2(*qiC[(ri - r) x Pil@l*G)& (16) where HeIis the electronic Hamiltonian To'+ V,energies and i
matrix elements are evaluated at the equilibrium nuclear geometry, and in eq 15 the parenthetic r indicates the length (position) version of the polar tensor. The length-velocity relation for the electronic electric dipole transition moment i
(7)
= -(im/h)(Wq
- WG)(*GlCril*q) i
(17)
which is a special case of the hypervirial relation (*Gl[xQi*ffll*q) i
Hansen. Aa. E.; Bouman, T. D. J . Chem. fhys. 1985, 82, 5035. Bouman, T. D.; Hansen, Aa. E. Chem. fhys. Le??.1988, 149, 510. Hansen, Aa. E.; Bouman, T. D. J . Chem. Phys. 1989, 91, 3552. Facelli, J. C.; Grant, D. M.; Bouman, T. D.; Hansen, Aa. E. J . Compur. Chem. 1990. 1 I , 32. (39) Bouman. T. D.; Hansen, Aa. E. In?. J . Quuntum Chem. Symp. 1989, 23, 381. (40)Nafie, L. A.; Walnut, T. H. Chem. fhys. Le#. 1977, 49, 441. (41) Nafie, L. A.; Polavarapu, P. L. J . Chem. fhys. 1981, 75, 2935. (42) Despite the prevalence of the term 'rotational strength" in the VCD literature. the term 'rotatory strength" is common (though not exclusive) in the electronic optical activity literature. Since optical activity measurements in the millimeter-wave and microwave regions of the spectrum are being discussed. (see, e.g., Polavarapu, P. L. J . Chem. fhys. 1987,86, 1136) one is faced either with adopting the term 'rotatory" or with the curious eventual prospect of discussing 'rotational rotational strengths"! We shall adopt the former course in this paper. (35) (36) (37) (38)
pilB
where the gauge origin l' simply serves as the reference point for the angular momentum operator. The nuclear contribution in eq 11 is similarly given as
(*GICpil*q)
contains both an electronic and a nuclear contribution
+
X
7
( o ~ ~ e l ), i@ =~(l h / 2 @ i ) ' / 2 ~ ~ A u . i p A u @
XA,
i
=
(wq
- wG)(*GlCQil*q)
(18)
xiQi,
for an arbitrary one-electron operator provides the velocity (momentum) alternative to eq 15, namely EA,,&) = - ( 2 i e h / m )
C (*GlaHcl/aXk,1*q)&* q*G ( v q
- vG)-2(*qICpi@(*G)& i
(19)
Equation 19 can be contracted toi5J6 EXu@@) =
-(2ieh / m ) (
( a ~ G ( R ) / ~ ~ , ) ~ I ( ~ * G ( ~ , ~ p ) / a(20) ~ 8 ) ~ ~ 0 )
where +G(&,As) is the electronic ground-state wave function at Ro in the presence of the perturbation
Theory of Vibrational Circular Dichroism
The Journal of Physical Chemistry, Vol. 95, No. 11. 1991 4257
- -
a m, b n. The matrices A and B are symmetric in the excitation indices. The atomic tensors fulfill the following sum r ~ l e s ’ ~ * ~ ’ * ~ ~ : The RPA method presupposes solving the Hartree-Fock equations to provide the sets of occupied and unoccupied molecular EEAap= -Ne&, (22) orbitals. Since, however, the RPA equations are invariant to A separate unitary transformations within each of these two sets of C ~ & ~ R O A*G ~ Ea ~p ~ 7(*~Iy7I*t(23) i)& orbitals, the orbitals can be chosen as canonical (delocalized) or Y noncanonical (e+, localized) with imp~nity.~’For generality 4hcCIm [ ~ ~ , ~ ( = r )-eCe=~7(*GlCri7l*G)~~ i (24) we therefore express the matrix elements of eqs 26-29 as A Y i
H’= (e”Ap(Cpip)i
(21)
where N is the number of electrons. Equations 22,24 and 23 are referred to as translational and rotational sums, respectively. For a # 8, both eqs 23 and 24 reproduce a component of the electronic part of the ground-state electric dipole moment. For CY = 8, eq 22 contains the Thomas-Reiche-Kuhn (TRK) sum rule for the electronic oscillator strength, while eqs 23 and 24 contain the Condon sum rule for electronic rotatory strengths, as discussed further in the Appendix. As presented, eqs 17, 18, and 22-24 assume exact wave functions and energies, and these relations may or may not be fulfilled in actual calculations. Violation of eq 17 makes the length and velocity versions of the atomic polar tensor, and hence R(U+l,r) and R(O+lg), different and makes R(O+l,r) dependent on the choice of gauge origin r, while R(O+l,p) remains gauge origin independent. The latter formal advantage of the velocity expression is counterbalanced by experience showing that considerable computational efforts are required to obtain numerically satisfactory results from R(O+l,p). On the other hand, the gauge origin dependence of the length expression is what offers the challenge of improving the accuracy with which R(O+l,r) can be calculated at a given level of computational effort through a judicious choice of gauge formulation. 111. Summary of the Random Phase Approximation The one-electron transition moments and energy differences appearing in eqs 15, 16, and 19 can be expressed, correct to first order in electron correlation, within the random phase approximati~n.’~,Restricting the discussion to molecules with a closed-shell electronic ground state, and noting that the operators in the expressions for the atomic tensors in section I1 are spin-free, we can limit the summations in eqs 15, 16, and 19 to spin-singlet excited states. The RPA expression for such a spin-singlet oneelectron transition moment is45 ( *GICQiI*q) i
am
[ ( hICQiIam)Xam,q + (amIX i
i
2”2C[(alQlm)Xam,q+ (mlQla)Yam,ql am
Yam,qI
(25a) (2%)
Here 14)is the Hartree-Fock single-determinant ground-state wave function, lam) is the spin-singlet wave function for the singly excited configuration generated by the orbital excitation a m, and a,b,c and m,n,k denote, respectively, occupied and unoccupied (virtual) molecular orbitals. Real orbitals will be assumed throughout. The amplitudes Xam,q are determined together with the energy differenceagGy5bq - WG from the coupled equations C ( A + B ) b n . a m ( X + Y)am,q = WqG(X - Y)bn,q (26)
-
am
Z ( A - B)bn,am(X - Y)am,q = WqG(X + Y)bn,q am
(27)
where the matrix elements are defined as Aam.bn = (amlHcl - W(44bn) (28) 4 m . h = (am,bnlH,llM (29) W ( 4 )is the Hartree-Fock ground-state energy and lam,bn) labels the spin-singlet wave function for the doubly-excited configuration (43) Laucretti, P. Adv. Chem. fhys. 1989, 75, 507. (44) Jsrgensen, P.; Simons, J. Second Quantization-Based Methods in Quanrum Chemistry; Academic Press: New York, 198 1. (45) Hansen, Aa. E.; Bouman, T.D. Ado. Chem. fhys. 1980, 44, 545. (46) Hansen, Aa. E.; Bouman, T. D. Mol. fhys. 1979, 37, 1713.
( A + B)am,bn = Fmn6ab
( A - B)am,bn
- Fab6,n Fmn6ab
+ 4(am)bn) - (bmlan) - (nmlab) - F a b s m n + (bmlan) - (nmbb)
(30) (3 1 )
where the Mulliken convention is used for the two-electron integrals, and F a b and F,,,,, are, respectively, matrix elements of the Fock operator within the occupied and unoccupied orbital sets. and Fab= &,, where e, and For canonical orbitals, F,,,,, = ,6“ ea are the orbital energies. The RPA amplitudes fulfill the following orthonormality and completeness relations:
(xp + Yq)t(X, - YP) = ,6
(32)
C(X, + YqNX, - YqY = 1 9
(33)
where X, and Y are column vectors labeled by singly excited configuration inlices am, and 1 is a unit matrix of order equal to the number of singly excited configurations included in the RPA calculation. From eqs 26, 27, 32, and 33 we obtain the following energy sum r ~ l e s : ~ ~ ~ ~ * C(X, + Yq)Wq~-’(Xq+ Yq)+= (A + B)-’ (34) 9
C(X, - Yq)Wq~-’(Xq - Yq)t = (A - B)-’ 9
C(Xq P
+ Yq)WqG-2(Xq- Y,,)’
= (A
+ B)-’(A - B)-’
(35) (36)
In eqs 32-36, q runs over all the (positive energy) solutions of eqs 26 and 27. Equations 34-36 are in fact in algebraic consequence of the linear equations (26 and 27) and hence hold regardless of basis set choice and choice of configurations lam) for a given RPA calculation. For exact solutions of the Hartree-Fock equations and of the associated RPA equations for a given system, the following relations (41[CQi,HeIIlam) = i
bn
[ ( 4lCQilbn )Abn,am - Bam.bn (bnlxail&) 1 (37a) i
i
and
(*GI [CQi,ffe11 W q ) = wqG(*GlCQil*q) i i
(37b)
These specialize to35949 (ih/m)(alplm) = C(blrln)(A - B)bn,am bn
(38a)
and (ih/m)(*GICiPiI*q) = WqG(*clCiril*q) (38b) since r is a real Hermitian operator. Thus, one-electron transition moments and excitation energies computed according to the RPA prescription of eqs 25-27 fulfill the hypervirial relations, eqs 17 and 18 and the sum rules in eqs 22-24 (see Appendix) in the Hartree-Fock limit. The Tamm-Dancoff approximation (TDA, identical with monoexcited CI) and the single transition approximation (STA) (47) Bouman, T. D.; Voigt, B.; Hansen, Aa. E. J . Am. Chem. Soc. 1979, 101, 550.
(48) Jsrgensen, P.;Oddenhede, J.; Beebe, N.H.F.J. Chem. fhys. 1978,
68, 2527.
(49) Harris, R. A.: Hansen, Aa. E.; Bouman, T. D. J . Chem. fhys. 1989,
91, 5856.
Hansen et al.
4258 The Journal of Physical Chemistry, Vol. 95. No. 11, 1991
are approximations to the RPA. TDA is obtained from eqs 26 and 27 by setting all B matrix elements and Y coefficients equal to zero, while STA further neglects all off-diagonal elements of the A matrix in the canonical basis. Neither STA nor TDA fulfills the hy rvirial relations and sum rules, even in the Hartree-Fock limit.,ge
IV. Atomic Polar and Axial Tensors With the machinery developed in the previous section, we can express the electronic tensors of section I1 in a number of forms that are convenient both for computational purposes and for the analysis of the resulting intensities. Noting that the derivative of the electronic Hamiltonian is the one-electron operator [aHeI/aHdR,, = -gzAT(rim
- p A a ) / l r i - Rod3a ZvAaVA(i) i
(39) Equations 15 and 19 for the electronic contribution to the atomic polar tensor become
and
(X- Y)t,,,,q(nlpBlb) ( 4 1 4 ( A - B)-lck,bn(nkplb) (41b)
from eqs 25b, 34, and 36. In eq 41b, an additional summation over the configurational indices is inserted to resolve the matrix product in eq 36. Equation 16 for the electronic contribution to the atomic axial tensor similarly becomes
PeW) = (e/mc) I3 CC(alVA,VAlm)(x + Y)am,qWqG-2' q#Gam bn (X- Y)bn,q(nl[(r - r) Plplb) (42a) = (e/mc)CCC(alo,,VAlm)(A am bn ck
+ @-'am,ck*
( A - @-'ck,bn(nl[(r
- r) x PI#)
(42b)
The structure of eqs 40b, 41b, and 42b suggests the introduction of three new transition moments, defined through the linear relations C(CIrAo,Ik)(A + @ck,bn = +ih ( ~ I V A , V A ~ ) (43) ck
C(cbplk)(A ck
C(clqf'lk)(A ck
- @ck.bn
- &,b
= (ih/m)(bb@ln) e
i(bl[(r - r) x Plpln)
(45)
A i
A i
*A
(47)
in complete orbital bases. If interpretations are insisted upon, eq 47 suggests that zAbe considered an atomic contribution to ~
~~
~~
(50) Harris, R. A. J . Chem. Phys. 1%9,50, 3947.
(48)
and EA,&) = -(4ie/h)CC(al*~lm)(ml~pla) a m
C@,p(P) (49) a
while the atomic axial tensor, eq 42b, becomes
IAas(r) = (e/hmc)CC(alrA,lm) (mlTbr)1a) 1Xik&') a m
a
(50) where the last step in eqs 48-50 defines contributions from doubly occupied orbital a to the respective atomic tensors. Since the present formulation is equally valid in canonical and noncanonical molecular orbital bases, we can adopt localized occupied orbitals to obtain decompositionsof the electronic parts of the atomic tensors into "bond" contributiomS2 By use of unprimed and primed orbital indices to denote localized and canonical bases, respectively, the explicit transformations of the EAa&) tensor contributions between the two bases can be expressed as (al*~Am)(mlrpla) = CCVam,vnt(b'lrdn')(k'lrplc') vm,am (51) where Vam,Vn,= Ttbtas,t and T and S are the unitary transformation matrices connecting localized and canonical orbitals in, respectively, the occupied and virtual orbital spaces." Since V is a unitary matrix, we find C(alrA,lm) (mlrpla) =
am
a'm'
(a'l*,lm')(m'lr,9Ja')
(52)
demonstrating the abovementioned invariance of eqs 48-50 with respect to choice of localized or canonical orbitals. Strictly analogous expressions obtain for the other tensor contributions. It follows further that the summation over the entire virtual space in the definition of the orbital contributions ekd and,"i makes it immaterial whether the virtual orbitals are localized. Returning to the total polar and axial tensors in eqs 7 and 11, we wish to treat the electronic and nuclear contributions on an equal footing, and since the electronic parts are expressed as sums over orbital contributioqs, we shall introduce the "orbital distributed nuclear charges" ZAa(Q)= ex"(Q)zA/eA(n)
(53)
where eA(W= I3ex"(Q) 2 f/3CCek',,(Q) a
(54)
8 - r
and for which obviously CzAa(Q) = ZA
(55)
a
Equations 7, 48, 49, and 53 then yield pA.p(Q)
it follows from eqs 37 and 39 that the matrix elements of defined in eq 43 fulfill the relation
C(Cl*x&) = (clp,(k) A
a
a m
(44)
With these definitions, the implied operators p and q(r) are real Hermitian, while is imaginary Hermitian. In the complete-basis limit, the matrix elements of p defined through eq 44 become equal to the matrix elements of the position operator r in accord with eq 38, and since [CPrHelI = -ihCCV,VA(i) = ihCCVAVA(i) (46) i
EAa@) = -(4ie/h)CC(a(rA.lm) (mlrpla) 1 Cek,&)
Vn'clr'
-(4ieh/m)CCC(a(V,,VAlm)(A + B)-lam,ck' am bn ck
the momentum operator. The operator #') implied by eq 4 9 ' has the general properties of an operator whose commutator with the electronic Hamiltonian is proportional to the angular momentum operator relative to the gauge origin at r; a coordinate representation of such an operator has apparently successfully evaded all search parties, In terms of the transition moments defined through eqs 43-45, the atomic polar tensors of eqs 40b and 4 1b become
=
= C[@,p(Q) + ez~~(QN,pl(56) a
~P".,~Q) a
providing an explicit orbital decomposition of the total atomic polar tensor in velocity (Q = p) or length (Q = r ) form. Similarly, the total atomic axial tensor can be written ~
~~~~~
(51) The operator ,cr) is the same as the effective operator discussed in ref 36 in the context of nuclear magnetic shieldings. (52) "Bond" = 'occupied. localized molecular orbital", which may be bonding or nonbonding.
The Journal of Physical Chemistry, Vol. 95, No. 11, 1991 4259
Theory of Vibrational Circular Dichroism a
= Z{ika8(r) + (ie/4hc)ZAa(Q)Cta@,1~AY - r,iJ (57) a
Y
V. Noncentrnl Gauge Origins: Dipole and Rotatory Strengths Equations 50 and 57 for IA&I‘)and @&r,n) contain a gauge origin, r, which so far remains unspecified. We now generalize eqs 50 and 57 by introducing nucleus- and orbitaldependent gauge origins Irk}and substituting the quantity ihQ@(r,rh,Q) = ika8(rk) + ( i / 4 h ~ ) Z : C t ~ , ~ ( I-’ ~rY)eka6(Q) , (58)
rk = Yx (all a) (68) where YAis an arbitrary vector associated with nucleus A. In a similar vein, orbital-based gauge choices need not adopt the orbital centroid choice of eq 65. Further, the dependence of Irk)on both A and a can be retained. For example, we could define
+
rb = (1 - tl)b~ tlpoa (0 Itl I1) (69) where 9 is an adjustable parameter. By use of eq 56, eq 3 for electric dipole transition moments becomes (OIP~I#I~)i
7 6
for ik&)
a
ha
( h / 2 4 ’ / 2 C P , @ ( Q )= (h/20,)’l2CP,#(Q) =
in eq 57. We then havem
P,(r,rk,n)
(h/2W,)”2CCS,~ka8(Q) = a,A
a
(h/2~i)’/~Cp~,rg(Q)(h/2~j)’/~pi,3(Q)(Q = r, p ) (70)
=
CikQ@(rk) + ( i / 4 h ~ )0C7C6C c @ , ~ (-r ~I’,)ekQd(Q) , (59) a and
MAQB(r,r”,n)= Cm”,(r,rk,n) = ZmbM(rk:Q) + ( i / 4 h a 1
A
Equation 1 for dipole strengths then yields D,(*l,Q,Q’) = (h/2W,) pk,(Q).pA’b,(n’)
c
a,bAA’
=
(h/2w,)Cpdi(Q).pbi(Q’) a,b
c ) ~ ~ ~ e-~r,)pka6(~) , ( ~ ,
(h /2w,)cPA,(Q)*PA’,(Q’)= (h/2@,)P,(Q)*P,(Q’) (72)
7 6
A X
(60)
providing length, velocity, and mixed length-velocity representations of dipole strengths depending upon the choice of Q and Q‘. If the occupied molecular orbitals are localized, the terms in eq 72 can be grouped into “bond” and “bond-bond” contributions; inspecting these can identify the orbitals contributing principally to the absorption intensity of the ith normal mode. By use of eq 60, eq 10 for magnetic dipole transition moments becomes (olgmaii ), = -(2h3w,)1/ZCCS,,m”,(r,r“,~) E
where mba8(rk,Q)= ikQ8(rh) + (ie/4hc)zA’(n)Ce~@~[~Ay - r”71 I
In the case of Q = p , eq 58 in fact equals ikQ8(r) (see eqs 44, 45, 49, and 50). and eq 60 merely provides an alternative decomposition of MA,#) to eq 57. When Q = r, on the other hand, MAq(I’)becomes a function of the choice of the set of gauge except in the Hartree-Fock limit. The set of gauge origins (P}, origins can thus be chosen to yield optimal accuracy of the MA, tensors, and hence of the resulting rotatory strengths. Two simple choices for the gauge origins (P} immediately present themselves. First, we can put
rk = R,O
(all a) (62) rk being the same for all orbitals a and located at the equilibrium position of nucleus A. In this case, eqs 59 and 60 become
a
ha
-(2h’~,)’/~C~”,~(r,r”,Q) = -( 2h 3~1)1/2Cma,s(r,r”,~) = h.
-(2 h 3
4 1/2CMA,@(r,rk,n)
-(2 n 3
a 4 l/2M,B(r,r“,n)
A
(Q = r, P) (73)
Equation 2 for rotatory strengths then yields R,(O+l,n,Q’) = h2 Im (CCpk,(Q).mA’b,(r,r”,Q’)~ = a.b.4.A‘
h2 Im {Cp”,(n).mb,(r,rk,Q’)} = h2 Im {P,(n).M,(r,r”,Q’)j
lAQ@(r,R:,Q) =
+ (i/4hC)CCe@$(RA; - ~ Y ) E ~ Q(63) ~(Q)
[’&Ao)
1
6
(74) If we define
MAa@(r,RA0,Q) = MAa@(RAo) + (i/4hc)CCe@76(RA: 7 6
- ~ Y ) F Q(64) ~(Q)
which are Stephens’ expressions for lA4 and MAdin the DO with origins at nuclei gauge.3 Secondly, we can associate the gauge origins Irk}with the orbitals a, independently of the nucleus A. If we locate the origins at the equilibrium centroids of the occupied orbitals a, then
rk = po
(alrla)o (all A)
(65)
mk,g(r”..Q) = CS,,mhsa,(rxl,Q)
(75)
Q
whence
m”ig(r,r”,n) = m”,@(rk,n)+ (i/4hC)C3t@76[rXl7 - r7iPh,6(Q) (76) 7.8
eq 74 can be expanded, giving R,(O+l,Q,Q’) = h2 Im (Ccpk,(a).[mA’b,(rA’b,S2’)
+
a.bA.A’
( i / 4 h c ) ( P - r) x pA’bi(n’)]}(77)
and eqs 59 and 60 become IAa@U‘@o,Q) =
Ei“~@b$) + (i/4hc)cCCe@76(pOa7- r7)eku6(Q)
(66)
a t 6
Mxa@(r@O,Q) =
Cmka@bOa)+ ( i / 4 h c ) ~ ~ C c @ & O a-7r7)Pku6(Q) (67) In the case that the molecular orbitals a are localized, this choice parallels the LORG approach of Hansen and Bouman for calculating nuclear shielding tensors.3s39 Alternative gauge choices exist, of course. A nucleus-based gauge more general than eq 62 is3
Equation 77 is dependent on the choice of gauge origin Q # Q’. However, when Q = Q’,eq 77 reduces to R,(O+l,n) = h2 Im (CCpk,(n).[mA~,(rA’b,n) +
when
a,bA,A’
(i14hc)rA’b x pA’b,(~)]}= h2 Im ZEp”,(Q).mA’b,(I’A’b,Q)+ a.bhA‘
(h/Sc)EEU‘” - ~A’b).[pL,(WX pA’i(Q)l (78) i.bhA‘
which contains no reference to the overall gauge origin I”. Equation 78 is simplified when the gauge origins {r”] are independent either of a or of A. Thus, for given by eq 62, eq 78 reduces to
4260 The Journal of Physical Chemistry, Vol. 95, No. 11, 19191 R,(O+l,Q) = h2 Im (X+,(Q).MA’,(R,P)}+ &A’
( ~ / ~ C ) ( C ( R-ARx8*[P(SOX, O X P(Q)”,Il (79) X.X’
which is Stephens’ equation in the DO with origins at nuclei gauge.’ Alternatively, for P given by eq 65, eq 78 reduces to RAO-.l,Q) = h2 Im
Ep’,(Q).mbibob,n)l + a.b
(n/8c)IEboa - PO~)*[P’,(Q) X pb,(Q)lI (80) a,b
which can be referred to as the LORG formula” for the vibrational rotatory strength. Thus,we have arrived at an expression for the rotatory strength, eq 78, which contains no reference to an overall gauge origin and which allows a flexible choice of noncentral gauge origins. This expression and both of its specialized versions in eqs 79 and 80 yield two representations, R,(O+l,r) and R,(O+l,p). In the case of the latter, the rotatory strengths are independent of the choice for any orbital basis, while except of the set of gauge origins (P) in the limit of a complete orbital basis, R(O+l,r) is dependent on this choice, allowing a search for noncentral gauge origins that are optimal for a given level of basis set quality.
VI. Discussion and Comparison with Previous Work We have derived expressions for atomic polar and axial tensors and for vibrational dipole and rotatory strengths. The essential points in the present derivation are as follows: (a) the use of the RPA formulation in the general form, eqs 25-3 1, allows noncanonical, and specifically localized, molecular orbital bases; (b) the energy sum rules, eqs 34-36, are used to remove the explicit sums-over-states in eqs 40-42, leading to the effective matrix elements defined by the linear equations (43-45) and hence to the (formally) simple orbital sums for the electronic atomic polar and axial tensors in eqs 48-50; (c) the introduction of noncentral gauge origins, eqs 58-61, leads to a general rotatory strength expression in eq 78 that contains only local quantities and internal distances and that specializes to Stephens’ DO expression? eq 79, and to a LORG expression, eq 80, analogous to Hansen and Bouman’s NMR shielding e x p r e s ~ i o n . ~ ~ The expressions for the rotatory strength, eqs 78-80, can be evaluated in either length (position) or velocity (momentum) representations. The velocity representations yield rotatory strengths that are independent of the choice of distributed origins, and the use of localized orbitals and noncentral origins serves only to produce decompositions of R,(O+l,p) into local terms, i.e., “bond” and “bond-bond” contributions;the accuracy of the results can be improved only through improving the orbital basis sets. On the other hand, as already emphasized, the length representation permits the choice of noncentral gauge origins that are optimal for a given level of computational sophistication. Since R,(O+l,r) is a signed quantity that varies linearly with the position of the gauge origins, the optimal location of these origins, strictly speaking, cannot be determined by an optimization procedure but must be arrived at from studies of convergence toward the complete basis set limit (where the origin choice is immaterial). It is important that the location of the origins can be obtained in a well-defined way from the molecular equilibrium structure; eqs 62, 65, and 69 all respect this requirement. The RPA was chosen as the point of departure not only because of the ease with which the option of localized molecular orbitals is introduced, but also to maintain the connection to the original presentation of the LORG method3sand to the RPA calculations of atomic tensors by Lazzeretti and co-workers.16-21-28 However, the resulting contracted expressions for the atomic polar and axial tensors in eqs 40b, 41b, and 42b can be obtained directly from the static limit of the appropriate polarization p r o p a g a t ~ r . ~It. ~ ~ follows that, apart from the implementational difference between calculating Exu&) (say) via eq 40a or 40b, the only difference (53) Oddershcde, J.; Jsrgensen, P.; Yacger, D. L. Compur. Phys. Rep. 1984, 2, 33.
Hansen et al. between the present approach and that of Lazzeretti and coworkers lies in introducing localized molecular orbital bases and the adjoint options of “bond” and “bond-bond” analyses and of flexible gauge origin choices. For a given basis and with untruncated RPA solutions, the approach of Lazzeretti and coworkers and the present approach will yield identical numerical results except when atomic axial tensors are evaluated by using orbital-based noncentral gauge origin choices and the length representation. The calculations of Lazzeretti and co-workers provide a clear indication of the dependence of the accuracies of length and velocity representations of atomic polar tensors on the choice of basis set. They have shown that results approaching Hartree-Fock limiting values require the use of large basis sets, unless specialized basis sets referred to as “polarized basis sets”’uu are employed, in which case EXd(r),but not EAd@), tensors are noticeably improved in accuracy. The improved accuracy of EAd(r)over Ex,-,&) when “polarized” basis sets are used implies that the accuracy of atomic axial tensors IAU,(I’,P,r)will be superior to that of IAa8(I’,I’h,p) for the various choices of noncentral gauge origins given in eqs 62, 65, and 69, and this expectation has been realized already in the case of eq 62.16.2’ It is to be expected that further improvements in accuracy in the calculation of ZAaa will be provided by optimizing the choice of both molecular orbital basis and gauge origins. The new orbital-based and other gauge choices discussed herein have been implemented in the RPAC Molecular Properties Package,” and numerical results will be presented in forthcoming publications. In the case of atomic polar tensors, the TDA and the STA (see section 111) have been compared with the RPA by Lazzeretti and co-workers in large basis set calculations for H20.*0 The TDA and STA were found to provide atomic polar tensors substantially lower in accuracy than those provided by the RPA. It is likely that similar conclusions will obtain the TDA and STA calculations of atomic axial tensors. Again, the present RPAC implementation will allow quantitation of this point. Direct implementation of eqs 8, 12, and 20 provides an alternative route for calculating atomic polar and axial tensor^.^'^ For a single determinant ground state OEC
EAa,dr)= -4eC ([~a(R)/a~~u]%Ir,lJ/a(Ro)) (8 la) a
EAu,@) = -(4ieh
/m)T(
/axAulR,,l
[a$s(RO,A@) /dAB]A,-O) (81b)
PUs(r)= 22([~a(R)/a~AulR,,l[a$~C$(RO,HB)/dH~l,,O) (81c) where +,(R), +&,AB), and +‘(&H8) are the occupied molecular and *o(&,HB). The derivatives orbitals of \ko(R), *&,A8), [ M R )lax~zl h,[Wi(RpA& la41ASO, and [Wa(W&laH~l tfpo can be obtained by using coupled Hartree-Fock perturbation theory. To date this has been carried out by using canonical molecular orbitals and atomic orbital basis sets {x,l that are nucleus-centered (and therefore a function of nuclear geometry, R) and independent of Hs and A For such basis sets, if we write +a = xpcapxp ( $ a ~ = xpcap$ at R = ~ 0 ) [Wa(R)/aX~ul%= C [(GP(R)/a&)hx,O + ca~(axp/a~Au)R,I(82a) P
[a$a(rb,A,8)/aABlA,=0= C(acap(rb,A,)/aAg)As-oXpO
(82b)
[Wa(rb,H~) /aHpIHpo = I3(acap(rb,H,)/aHg)Hp=OXpO
(82~)
P
P
coupled Hartree-Fock perturbation theory expands the coefficient (54) human, T. D.; Hansen, Aa. E. RPAC, version 9.0; Southern Illinois University at Edwardsville: Edwardsville, IL, 1990.
The Journal of Physical Chemistry, Vol. 95, No. 11, 1991 4261
Theory of Vibrational Circular Dichroism derivatives of eqs 82 in terms of unperturbed coefficients:
(Gp/aA)o = XUAam@mp
(83)
m
where A denotes the relevant perturbation parameter (&, H5, or A,) and provides equations for the matrix UA,the first-order CHF equations. Equations 81-83 lead to EA,,(r) =
-4e[CiC v x b , m (hO1rfl+ l$ (:)) a
m
1
/axAct)&,lr#:)
(84a)
is of interest to ask whether these formulas can be derived by well-defined approximations to the rigorous, general expressions reviewed herein. In several cases this has been shown to be One such formula, resulting from the so-called localized molecular orbital (LMO) m ~ d e I , ~has * ~not ' yet been analyzed in this way. The introduction of localized molecular orbitals and of local gauge origins for individual molecular orbitals in this paper provides a basis for such an analysis. Taking eqs 23 and 24 of ref 41 as the basic working equations of the LMO model and defining the magnetic and electric dipole transition moments, respectively, it is easily shown that the LMO model implies an atomic axial tensor with electronic part
EA,,(r) =
-(4ieh /m)CCIUxbamUApma+ ( (a'h/ax,),toI+mo)
~ ~ ~ =, ((i/4hc)CCC4&$., r)
uApmnl
- r.,)@,,
(87)
a m
(84b)
(85) The equations determining the Ux and U Hmatrices are given by eqs 14-19 of ref 9, and UAis obtained by appropriate modification of eq 17 of that reference (see also ref 15). Comparison of eqs 84 and 48-50 assuming canonical molecular orbitals in the latter, shows that these sets of equations are formally identical if we equate Uxbam uA6ma
UH6,,,, =
(i/h)(al*dm)
(86a)
= (-im/h) (mIp&)
(86b)
(-ie/2m~)(mlt$~)la)
(86c)
and adopt an atomic basis {x,o)centered at the equilibrium nuclear positions so that (a$,/aXh), = 0. Further, it follows that the CHF equations for UA( A = A , H5)9are identical with eqs 44 and 45, while the CHF equation9 for Uxh becomes identical with eq 43 when the atomic orbital derivatives (d~,/dX,)~are zero. Thus, eqs 43-45 and 48-50 replicate the equations that would be obtained from coupled Hartree-Fock perturbation theory if the atomic orbital basis set is taken to be independent of nuclear coordinates, R. In the limit of a complete basis set, the RPA and CHF approaches will generate identical results; finite basis sets will of course yield different results. Since the present approach is equivalent to the analytical derivative CHF approach with a less flexible basis set, it is to be expected that (for comparable accuracy) it will require larger basis sets. The introduction of "polarized" basis sets into RPA calculations of atomic polar and axial tensors is designed to fulfill efficiently the need for such basis sets. While the requirement for larger basis sets is a disadvantage, it is compensated by a substantial diminution in computational labor, since integrals involving basis function derivatives (ax / ax,), are eliminated, and in practice the balance may well {e determined by the actual implementation of the methods and the available computer power. Although analytical derivative CHF methods using noncanonical orbitals have been formulated and applied to the calculation of magnetic proper tie^,"-^ to date there has been no application to atomic polar and axial tensors. The improvements obtainable from a search for optimal gauge origin choices of the more general form available in the present approach remain to be explored. A number of approximate formulas for magnetic dipole transition moments have been put forward and used in the calculation -~' of rotatory strengths and the analysis of VCD ~ p e c t r a . ~ ~ It ~~~
~
~
~~
in our notation. Comparing eq 87 with eq 66 shows that the LMO model equation for vibrational magnetic dipole transition moments and rotatory strengths is thus the approximation in which all ih,5(poa) tensors are assumed to be zero. From eqs 50 and 45, this amounts to neglecting all matrix elements of the form
( mIT@0')&3)= iC(A - B)-'am,bn(nI[(r - PO') X PI#') bn
(88)
Hence, the present, more complete theory shows that the essential approximation in the LMO approach lies in the neglect of local transition moments of the magnetic operator 7 . In the angular momentum representation of the magnetic dipole moment operator, it follows from the right side of eq 88 that a number of nonlocal transition moments are neglected in the LMO model, in contrast to the assumptions stated in refs 40 and 41. As with the Atomic Polar Tensor, Fixed Partial Charge, and Coupled Oscillator e q u a t i o n ~ , the ~ J ~LMO equation thus omits a part of the physics of vibrational magnetic dipole transition moments. One consequence of this is that, even if implemented at the ab initio SCF level of approximation, the LMO model does not yield results for Pa, tensors that converge to the Hartree-Fock limit as the basis set approaches completeness. Whether or not the missing terms are numerically crucial in the finite basis set calculations is of course open to experimentation, and our RPAC implementation of the ika5(~oa)tensors provides an opportunity to shed light on the question. Calculations of atomic polar and axial tensors and of vibrational dipole and rotatory strengths using sum-over-states expressons and the STA have been published very recently for EA,, and IAue by Rauk and co-worker~.~~ In this work, eqs 15, 16, and 19 are rewritten in the forms EAu&) =
-2e
c ((a\kG(Ro)/ax,),l\kq(%)
1(*q(Ro)lCri,l'ko(Ro) 1
q+G
)
(89a) EA,,&) = (2ieh/m)
((~~~(Ro)/ax~~)~~~q(Ro)). q+G
( w q
- w~)-'(\kq(Ro)IC~i,l\k~(Ro)) (89b) 1
I A u g ( r ) = -(e/2mc)
((a\kG(%)/axAa),l\k.9(Ro))' q#G
(pq - w G ) - ' ( ~ q ( % ) l ~ [ ( r i - r) x p 1 I a ~ \ k G ( ~ () )8 9 ~ ) i
The sums over excitations in eqs 89 are implemented by using the STA; Le., the \k,(&) are obtained by single orbital promotions from the SCF ground state e,(&)and the excitatfin-energies ( - wG)exclude effects of configuration mixing. The derivative wavefunctions (a\ko(&)/aX,), are evaluated analytically by using the CHF equations. It would appear that this approach suffers from the principal disadvantages of both the RPA approach and the analytical derivative CHF approach, requiring both "polarized" basis sets and, in addition, the calculation of integrals
wq
~
(55) Nafie, L. A. Adu. Infrared Raman Spectrosc. 1984, I ! , 49. (56) Stephens, P. J.; Lowe, M. A. Annu. Reu. Phys. Chem. 1985,36,213. (57) Nafie, L. A,; Freedman, T. B. In Topics in Stereochemistry; Eliel, E. L., Wilen, S. H., Eds.; Wiley: New York, 1987; Vol. 17, p 113.
(58) Dutler, R. Ph.D. Thesis, University of Calgary, 1988. Dutler, R.; Rauk. A. J . Am. Chem. Soc. 1989,111,6957. Rauk, A.; Dutler, R.; Yang, D. Can. J. Chem. 1990, 68, 258.
Hansen et al.
4262 The Journal of Physical Chemistry, Vol. 95, No. 11, 1991 involving atomic orbital derivatives. Further, no matter what the basis set, it suffers from the deficiencies inherent in the STA and does not converge with increasing basis set size to the HartreeFack limit.
VII. Conelusioa We have formulated expressions for atomic polar and axial tensors allowing the use of localized molecular orbitals and of individual gauge origins for individual orbitals. These expressions have been derived via the RPA and are rigorous at the SCF level of approximation; i.e., they yield Hartree-Fack limiting results in the limit of complete orbital basis sets. Localized molecular orbitals and individual gauge origins for individual orbitals have been used extensively in ab initio SCF calculations of nuclear magnetic shielding tensors and magnetic susceptibilities. Our work provides a basis for their use in calculating atomic polar and axial tensors, vibrational rotatory strengths, and VCD spectra. The methods described herein have been implemented in the RPAC program of Bouman and Hansen. Calculations will permit the dependence of the accuracy of predictions on choices of basis set, noncanonical orbitals, and gauge origins to be defined. In addition, the utility of such approximations as the TDA and STA and of the equations resulting from the LMO model will be more precisely quantitated. Acknowledgment. Aa.E.H. thanks the Danish Carlsberg Foundation for a grant in support of this work, P.J.S. acknowledges support from the NSF, the NIH, and NATO, and T.D.B. acknowledges support from the NSF (Grant CHE-8610413). Appendix We shall demonstrate here that the expressions for the atomic polar and axial tensors given in section I11 fulfill the sum rules in eqs 20-22 in the limit of a complete basis set. For the translational polar sum rule, eq 20, eqs 46 and 45 yield
ZEA.g = -4(ie/h)Z (alp.lm) (mlrg(a) A
am
(A. 1)
b
(A.4)
and subsequently to use the relation
ZZRoA [vAvA(i)] = Xri A I
i
[vjvA(i)]= (i/h)[C(r, x PI), H e 1 1 64.5) i
the first part of which follows from Newtonian mechanics, together with eq 35, to obtain
ZZf#-#A@Ak= A 76 W e / h)C ( 4 I U r i x pJglam) (amlCr,(i)l&) am
I
1
= ( 4 i e / h E ( a I ( r x P)slm)(mlr,la) am
= 2ie/hC(a([(r
X
a
p ) ~ r,lla) , = 2eCt,g,C(a(r7(a) (A.6)
where we use again eq A.2 and the vanishing of all terms arising from the double summation over a and b. Thii completes the p m f of the rotational axial sum rule, since the final expression in eq A.6 represents the dipole moment of the electronic ground state as given by the HartreeFack wave function, which is known to be the consistent ground-state function to use when the RPA or CHF is used for the response tensors. For the translational axial sum, eq 22, eqs 40b, 41,45, and 42 combine to yield 4hcCIm {Pa,&')/ = A
-4e/h Im ~ Z ( a l ~ , I m ) ( m l ~-(r) r x ~ l & d= am
Nl2
CEAag = - 2 ( i e / h ) c (aIl[p,,r@llaI)= -Ne&,
-2eZ.c
(A4
and utilizing that all resulting terms involving the double summation over a and b cancel identically, eq A.l becomes 1-1
1
a
Clm)(ml = 1 - Clb)(bl
A
( A + B)-'a,.a(bnlCr,(i)l&o)
-2e/h Im ( C ( a l [ r J ( r - r) x PIg1la)l =
Inserting the completeness relation m
and that it reduces to a sum over orbital excitations in the mixed velocity-length formalism.59 For the rotatory polar sum rule, eq 21, it will be convenient to utilize eqs 23 and 28 to write
(A.3)
as was to be shown. Since the electronic oscillator strength in the mixed velocity-length form is given as fos = (2i/3h)(Glplq)*(qlrlG),eqs A.l-A.3 become a restatement of the proof that the TRK sum rule holds in the RPA in complete orbital basesSo
7
C(al(r,
- r,)la)
(A.7)
where the right side is a component of the electronic ground-state dipole moment relative to an origin at I', as was to be shown. Equations A.6 and A.7 are both seen to contain the Condon sum rule for the electronic rotatory strength in the length form for ci = 8. (59) Hansen, Aa. E.; Bouman, T. D. Chem. Phys. Lcrr. 1977, 45, 326. (60) The dependence on indicated on the left sides of eqs 59 and 60 ia intended to emphasize that the numerical reaults may depend upon the ser of gauge origins rb.
rb
