- Email: [email protected]
Diagrammatic CASPT2 using an internally contracted basis
18 February 2000
Chemical Physics Letters 318 Ž2000. 190–195 www.elsevier.nlrlocatercplett
Diagrammatic CASPT2 using an internally contracted basis James P. Finley a
a,b,)
Department of Applied Chemistry, Graduate School of Engineering, UniÕersity of Tokyo, Tokyo 113-8656, Japan b Department of Theoretical Chemistry, Chemical Centre, P.O. Box 124, S-221 00 Lund, Sweden Received 8 November 1999; in final form 3 December 1999
Abstract The diagrammatic CASPT2 method is derived for an internally contracted basis. The wave-operator is efficient to compute and does not require orthogonalizing the internally contracted basis set and diagonalizing the zeroth-order Hamiltonian H0 . The wave-operator also has a simple form that is a generalization of Møller–Plesset perturbation theory. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The MRMP w1,2x and CASPT2 w3,4x methods are two ab initio methods that have been successful in computing electronic spectra of medium-sized molecules. However, these methods have their shortcomings. For example, the first-order wave-operator for MRMP has a different form for each determinantal state upon which it acts. This ket dependence makes the MRMP method less efficient, since the final energy is computed as a function of the transition density matrices involving the CASSCF state of interest < a : and each of the determinantal states within the CAS. The CASPT2 method avoids this shortcoming by using a non-orthogonal, internally contracted basis. Unfortunately, before computing the first-order wavefunction, CASPT2 requires steps to orthogonalize this non-orthogonal basis and diago-
nalize H0 , partially offsetting the advantage gained from using an internally contracted basis. The diagrammatic CASPT2 method w5x has a much more efficient formalism than either MRMP or CASPT2 but is designed to yields similar results. Recently the accuracy of diagrammatic CASPT2 has been demonstrated by computations of N2 , benzene and LiF w6x. Below we show that the first-order wave-operator
2. Diagrammatic CASPT2 theory
)
Fax: q81-3-5841-7241; e-mail: [email protected]
The first-order wave-operator V Ž1. is required to compute the second-order energy E Ž2.. This operator
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 1 4 4 7 - 5
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
generates the first-order wavefunction
V
Ž1. <
a : s
Ž 1.
where < a : is the CASSCF Žor CASCI. state of interest, < a : s Ý Cp < p: ,
Ž 2.
p
and < p : denotes the pth determinantal state from the CAS. The first-order wave-operator for MRMP, denoted Ž1. by V mp , is ket dependent, Ž1. Ž p. < : ² < V mp s Ý V mp p p.
The pth component of this wave-operator is given by w9x Ž p. V mp sÝ
wr
q 12
w r < h0 < w x † a a ´ˆ wr q D p r w w rw < sx x
Ý ´ˆ r s q D a†r a†s a x a w , p rswx w x
where the zeroth-order energies are given by Ei0 s ² i < H0 < i : .
Ž 4.
where
w r < h0 < w x † ar aw ´ˆ wr q Dˆwr
V dŽ1. s Ý wr
q 12
Ý rswx
´ˆ wr s eˆw y eˆr ,
Ž 5.
sx
´ˆ wr sx s eˆw q eˆ x y eˆr y eˆs ,
Ž 6.
q 12
Ž 7.
w ij < kl x are two-electron integrals written in chemist’s notation w10x; w and x denote inactive and active orbitals; while, r and s denote active and secondary orbitals, except that all indices cannot simultaneously be active Žno internal excitations.. Furthermore, the orbital energies eˆi define a diagonal, one-body, zeroth-order Hamiltonian,
Ž 8.
i Ž1. The ket dependence of V mp is caused by the shifts in the energy denominator D p that can can be interpreted as shifts that force the < p : states to be degenerate w9x,
D p s Ea0 y Ep0
,
Ž 9.
Ž 11 .
w s < h0 < x x s < aˆ x : ´ˆ xs q Dˆ xs
Ý rswx
a
H0 s Ý eˆi a†i a i .
w rw < sx x † † ar as a x aw , ´ˆ wr sx q Dˆwr sx
where the shifts, Dˆwr and Dˆwr sx , are at our disposal w11x, but are chosen in a manner so that the wave-operator V dŽ1. for diagrammatic CASPT2 is similar to the one used in the MRMP method, given by Eqs. Ž3. and Ž4.. Diagrammatic CASPT2 can also express its firstorder wave-function
i < h 0 < j s w i < h < j x q Ý Ž w ij < aa x y w ia < aj x . ,
Ž 10 .
The first-order wave-operator for diagrammatic Ž1. , CASPT2, denoted by V dŽ1., has the same for as V mp Ž . given by Eq. 4 , except that one- and two-body shifts, Dˆwr and Dˆwr sx , replace the ket dependent shift D p w5x,
Ž 3.
p
191
w rw < sx x < aˆ wr sx : , ´ˆ wr sx q Dˆwr sx
Ž 12 .
where the internally contracted states are generated by applying excitation operators to < a :, < aˆ xs : s a†s a x < a : ,
Ž 13 .
< aˆ wr sx : s a†r a†s a x a w < a : .
Ž 14 .
3. Energy denominators The one-body shift Dˆwr is given as a weighted average of the shifts D p for pth determinantal states < p : that can undergo w r excitations. Explicitly, they are given by
™
Dˆwr s
1 Nwr
Ý nŽwp. Ž 1 y nŽr p. . D p < C p < 2 , p
Ž 15 .
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
192
where
The following identities are proved in Appendix
Nwr s Ý nŽwp. Ž 1 y nŽr p. . < C p < 2 ;
Ž 16 .
p
C p2
the squared coefficient is the weight of < p : in the CASSCF state of interest, given by Eq. Ž2., and nŽi p. is the occupation – 0 or 1 – of the ith orbital in < p :. Similarly, the two-body shifts, that correspond to double excitations, are given by 1
ˆ
Dwr sx s
Nwr xs
Ý
nŽwp. nŽxp.
Ž
1 y nŽr p.
.Ž
1 y nŽs p.
. Dp < Cp <
2
1
Ý nŽwp. nŽmp. Ž 1 y nŽr p. . D p < C p < 2 , N rm
Ý
´ˆ wr sx q Dˆwr sx s Ea0 y E˜raw0s x ,
Ž 26 .
where E˜raw0 and E˜raw0s x are zeroth-order energies, E˜raw0 s
² aˆ wr < H0 < aˆ wr : ² aˆ wr < aˆ wr :
Ž
1 y nŽr p.
. < Cp < 2 ,
Ž 27 .
² aˆ wr sx < H0 < aˆ wr sx :
E˜raw0s x s
Ž 18 .
of the normalized, internally contracted states, given by Eqs. Ž13. and Ž14.. Using the identities, given by Eqs. Ž25. and Ž26., Eq. Ž12. can be written as
² aˆ wr sx < aˆ wr sx :
,
Ž 28 .
w s < h0 < x x s < aˆ x : Ea0 y E˜sax0
Ž 19 .
p
nŽwp. nŽmp.
,
Ž 17 .
The two-body shift correspond to single excitations whenever either w or x is an active orbital, say m, and either r or s is the same active orbital. These types of excitations are defined by
Nwr xs s
Ž 25 .
,
p
wm
´ˆ wr q Dˆwr s Ea0 y E˜raw0 ,
p
Nwr xs s Ý nŽwp. nŽxp. Ž 1 y nŽr p. .Ž 1 y nŽs p. . < C p < 2 .
Dˆwr mm s
A:
q 12
Ý rswx
Ž 20 .
w rw < sx x < aˆ wr sx : . Ea0 y E˜raw0s x
Ž 29 .
p
where all two-body shifts corresponding to single excitations are equal, mr rm Dˆmw s Dˆmw s Dˆwmmr s Dˆwr mm .
Ž 21 .
Eqs. Ž17. and Ž19. can be combined into a single relation that defines all two-body shifts,
Dˆwr sx s
1 Nwr xs =Ž
Ý
Using a restricted set of spin orbitals, the waveoperator for diagrammatic CASPT2 can be written as
V dŽ1. s Ý
Gr nŽwp. nŽxp. 1 y nŽr p. w x
Ž
4. Restricted formalism
.
wr
Ž r < h0 < w . ´ wr q Dwr
Eˆr w q
Ž rw < sx .
1 2
Ý rswx
´ wr sx q Dwr sx
Eˆr w s x ,
p
Ž 30 .
Gs 1 y nŽs p. w x D p < C p < 2
.
Nwr xs s Ý nŽwp. nŽxp. Ž 1 y nŽr p. .
G wr x
,
Ž 1 y nŽs p. .
G ws x
Ž 22 .
where
2
´ wr s e w y e r ,
Ž 31 .
´ wr sx s e w q e x y e r y e s ,
Ž 32 .
< Cp < ,
p
Ž 23 .
Ž i < h0 < j . s Ž i < h < j . q Ý 2 Ž ij < aa . y Ž ia < aj .
where
,
a
Ž 24 .
Ž 33 .
Note that terms involving Ž Gwix s y1. are not considered, since w s x terms do not occur in the first-order wave-operator, given by Eq. Ž12..
and Ž ij < kl . are spin independent two-electron integrals written in chemist’s notation w10x. Henceforth, the indices w, x, r, and s refer to spatial orbitals;
Gwix s 1 y d i w y d i x .
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
spin orbitals are denoted by ws , x s , rs , and ss , where s s a or b . The spin-adapted excitation operators w12x, or unitary group generators, given in Eqs. Ž34. and Ž35.,
s
Ý a†rs a†ss s
1
Ý N rs nŽwp.s Ž 1 y nŽrsp. . D p < C p < 2 ,
Dwr s 12 Ý Dwr sx s 14
X
X
a x s X a ws ,
Ž 35 .
1
Ý Ý N rs ss
ss X p
H0 s Ý e i Eˆi i ,
Ž 36 .
= Ž 1 y nŽs sp.X .
G ws ss x s X
Dp < Cp < 2 ,
where e i s e i s ; therefore, the following identities are valid and independent of s and s X : ,
Ž 46 .
p Ž p. Nwrss xs ss X s Ý nŽwp.s nŽxp. s X Ž1 y n rs .
X
Ž 38 .
By substituting Ž30. into Eq. Ž1. we get an expression for the first-order wave-function,
q 12
Ž s < h0 < x . ´ xs q D xs
< a xs :
Ž rw < sx .
Ý
´ wr sx q Dwr sx
rswx
< a wr sx : ,
Ž 39 .
X
= Ž 1 y nŽs sp.X .
ˆ
< a xs : s Es x < a : ,
Ž 40 .
< a wr sx : s Eˆr w s x < a : .
Ž 41 .
A reasonable, spin-adapted, modification of Eqs. Ž15. and Ž22. is given by an average over all possible spin combinations,
Dwr s 12 Ý Dˆwr ss , X
,
Ž 43 .
are given by Eqs. Ž15. and Ž22., with r, w, s, x,
Ž 48 .
X
Ž 49 .
By adding together the equations involving all spin states and using Eqs. Ž27., Ž28., Ž37., Ž38., Ž42. and Ž43. we get the following identities:
´ wr q Dwr s Ea0 y Eraw0 ,
Ž 50 .
´ wr sx q Dwr sx s Ea0 y Eraw0s x ,
Ž 51 .
where Eraw0 and Eraw0s x are average zeroth-order energies, Esax0 s 12 Ý
² aˆ xsss < H0 < aˆ xsss :
s
ss X
X where the spin-dependent shifts, Dˆwr ss and Dˆwr ss sxss X ,
Ž 47 .
´ˆ wr ss sxss X q Dˆwr ss sxss X s Ea0 y E˜ras0ws s s X x s X .
Ž 42 . X
< Cp < 2 ,
´ˆ wr ss q Dˆwr ss s Ea0 y E˜ras0ws ,
s
Ý Dˆwrss sxss
G ws ss x s X
and nŽi sp. is the occupation – 0 or 1 – of the ith orbital in < p : with s spin. Eqs. Ž44. and Ž45. differ slightly from the previous formulation of diagrammatic CASPT2 w5,6x. ŽNote that the equations for the energy denominator shifts in Ref. w5x have errors that are pointed out in Ref. w6x.. By substituting rs , ws , ss X , x s X for r, w, s, x, respectively, Eqs. Ž25. and Ž26. become
X
where the internally contracted states are generated by applying the excitation operator to < a :,
G wrss x s X
p
Ž 37 .
´ wr sx s ´ˆ wr ss sxss X .
Ž 45 .
Nwrss s Ý nŽwp.s Ž 1 y nŽr sp. . < C p < 2 , X
ˆ
G wrss x s X
where the weights are
i
Dwr sx s 14
ws x s
Ž p. nŽwp.s nŽxp. s X Ž1 y n rs .
X X
X
insure that our formalism is spin-adapted. Furthermore, the orbital energies e i are eigenfunctions of a diagonal one-body zeroth-order Hamiltonian,
´ wr s ´ wr ss
Ž 44 .
ws
p
Ž 34 .
s
Eˆr w s x s Ý
replaced by rs , ws , ss X , x s X , respectively. Explicitly, these equations are given by
s
Eˆr w s Ý a†r s a ws ,
193
² aˆ xsss < aˆ xsss :
,
Ž 52 .
X
Eraw0s x s 14
Ý
ss X
X
² aˆ wr ss sxss X < H0 < aˆ wr ss sxss X : X
X
² aˆ wr ss sxss X < aˆ wr ss sxss X :
,
Ž 53 .
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
194
of the normalized, spin-dependent, internally contracted states, given by < aˆ xsss : s a†s s a x s < a : , X
< a wr ss sxss X : s a†r s
ˆ
Ž 54 .
a†s s X a x s X a ws
Ž 55 .
As an alternative to Eqs. Ž52. and Ž53., Esax0 and a0 Er w s x can be taken as a generalization of Eqs. Ž27. and Ž28., Esax0 s
² a xs < H0 < a xs : ² a xs < a xs :
Eraw0s x s
,
Ž 56 .
² a wr sx < H0 < a wr sx : ² a wr sx < a wr sx :
,
Ž 57 .
where the internally contracted states are given by Eqs. Ž40. and Ž41.. The corresponding one- and two-body shifts, Dwr and Dwr sx , can then be computed using Eqs. Ž50. and Ž51.. Using the identities Žor definitions., given by Eqs. Ž50. and Ž51., Eq. Ž39. can be written as
Ž s < h0 < x .
Ea0 y Esax0
sx
q 12
rswx
Ea0 y Eraw0s x
< a wr sx : .
Ž 58 .
When one or more active electrons are present, we can express the first-order wave-operator in a form with the single excitations combined with the double excitations,
5. Comparison with single-reference perturbation theory Our formalism reduces to second-order Møller– Plesset perturbation theory when there are no active orbitals. For a single-determinantal reference-state, denoted by < a:, the first-order wave-operator is given by Eq. Ž1. with < a: replacing < a :,
V Ž1. < a: s
Ý Cwr sx < a˜ wr sx : ,
Ž 59 .
Ž 62 .
where the first-order wave-operator,
q 12
w s < h0 < x x s < aˆ x : Ea0 y E˜sa0x
Ý rswx
< a xs :
Ž rw < sx .
Ý
set without orthogonalizing these basis functions and diagonalizing H0 .
w rw < sx x r s < aˆ w x : , Ea0 y E˜ra0w s x
Ž 63 .
is identical to Eq. Ž29., except that the Žzeroth-order. excited states are generated using the single determinantal reference state, < aˆ sx : s a†s a x < a: ,
Ž 64 .
< aˆ wr sx : s a†r a†s a x a w < a: .
Ž 65 .
Furthermore, the zeroth-order energies are given by the expectation value of the zeroth-order Hamiltonian,
rwsx
where the internally contracted states < a˜ wr sx : and coefficients Cwr sx are given by
ˆ Eˆs x < a : ,
< a wr sx : s Er w
˜
Cwr sx s
Ž rw < sx . Ea0 y Eraw0s x yÝ m
q
Ž 60 . dr w
2 Ž s < h0 < x .
nÕ
Ea0 y Esax0
Ž sm < mx . a0 Ea0 y Esm mx
,
Ž 61 .
and the summation over m includes only active orbitals. Hence, unlike CASPT2, the first-order wave-operator
E˜sa0x s ² aˆ sx < H0 < aˆ sx : ,
Ž 66 .
E˜ra0w s x s ² aˆ wr sx < H0 < aˆ wr sx : ,
Ž 67 .
and Ea0 is given by Eq. Ž10.. Eqs. Ž27. and Ž28. are generalizations of Eqs. Ž66. and Ž67.. The additional denominator factors for Eqs. Ž66. and Ž67. seem appropriate, since the internally contracted states, < aˆ xs : and < aˆ wr sx :, are not normalized. For a one-body zeroth-order Hamiltonian, the energy denominators can be written as Ea0 y E˜sa0x s ´ˆ wr ,
Ž 68 .
Ea0 y E˜ra0w s x s ´ˆ wr sx ,
Ž 69 .
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
where ´ˆ wr and ´ˆ wr sx are given by Eqs. Ž5. and Ž6.. These equations are identical to Ž25. and Ž26., except for the absence of the one- and two-body shifts, Dˆwr and Dˆwr sx . These shifts only depend on active indices, and vanish when no orbitals are active. Because of Brillouin’s theorem, the term corresponding to single excitations in Eq. Ž63. vanishes. Similarly, for a CASSCF reference state < a :, single excitations vanishes when x is inactive and s is secondary w13x.
the numerator factors in Eqs. Ž27. and Ž28. are also easily evaluated, ² aˆ wr < H0 < aˆ wr : s Ž eˆr y eˆw . ² aˆ wr < aˆ wr : q Ý nŽwp. Ž 1 y nŽr p. . < C p < 2 Ep0 , p
Ž A.5 . ² aˆ wr sx < H0 < aˆ wr sx : s Ý nŽwp. nŽxp. Ž 1 y nŽr p. .
G wr x
p
= Ž 1 y nŽs p. .
Acknowledgements
˚ Malmqvist The author would like to thank P.-A. for useful discussions.
195
G ws x
< C p < 2 Ep0
y ´ˆ wr sx² aˆ wr sx < aˆ wr sx : .
Ž A.6 .
Appendix A. Proofs for Eqs. (25) and (26) identities
By substituting Eqs. Ž15. and Ž27. into Ž25., and using ŽA.1., ŽA.5., Ž5., and Ž9., the Eq. Ž25. identity is proved. By substituting Eqs. Ž22. and Ž28. into Ž26., and using ŽA.2., ŽA.6., Ž6., and Ž9., the Eq. Ž25. identity is proved.
Using Eqs. Ž13. and Ž14., the denominators in Eqs. Ž27. and Ž28. are easily evaluated, yielding
References
² a wr < a wr : s Nwr
ˆ ˆ
,
Ž A.1 .
² aˆ wr sx < aˆ wr sx : s Nwr xs ,
Ž A.2 .
where Nwr and Nwr xs are given by Eqs. Ž16. and Ž23.. Since internal excitations are not considered, Ž w / r . for Eq. Ž25.. Also, Ž w / x . and Ž r / s . cases are not considered for Eq. Ž26., since these terms are zero for the first-order wave-operator: they violate the Pauli exclusion principle. Using the identities given by a†r a w H0ˆ < p : s a†r a w < p : Ž eˆr y eˆw q Ep0 . ,
Ž A.3 .
a†r a†s a x a w H0ˆ < p : s a†r a†s a x a w < p : eˆr q eˆs y eˆw y eˆ x q Ep0 ,
ž
/
Ž A.4 .
w1x K. Hirao, Chem. Phys. Lett. 190 Ž1992. 374. w2x K. Hirao, Chem. Phys. Lett. 196 Ž1992. 397. ˚ Malmqvist, B.O. Roos, A.J. Sadlej, K. w3x K. Andersson, P.-A. Wolinski, J. Phys. Chem. 94 Ž1990. 5483. ˚ Malmqvist, B.O. Roos, J. Phys. Chem. w4x K. Andersson, P.-A. 96 Ž1992. 1218. w5x J.P. Finley, J. Chem. Phys. 108 Ž1998. 1081. w6x J.P. Finley, H.A. Witek, J. Chem. Phys. in press. w7x R.J. Bartlett, D.M. Silver, J. Chem. Phys. 62 Ž1975. 325. w8x J.A. Pople, J.S. Binkley, R. Seeger, Int. J. Quantum Chem. 10S Ž1976. 1. w9x J.P. Finley, Chem. Phys. Lett. 283 Ž1998. 277. w10x A. Szabo, N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, 1st edn., Macmillian, New York, 1982. w11x J.P. Finley, J. Chem. Phys. 109 Ž1998. 7725. w12x B.O. Roos, P. Linse, P.E.M. Siegbahn, Chem. Phys. 66 Ž1981. 197. w13x B. Levy, G. Berthier, Int. J. Quantum Chem. 2 Ž1968. 307.
Chemical Physics Letters 318 Ž2000. 190–195 www.elsevier.nlrlocatercplett
Diagrammatic CASPT2 using an internally contracted basis James P. Finley a
a,b,)
Department of Applied Chemistry, Graduate School of Engineering, UniÕersity of Tokyo, Tokyo 113-8656, Japan b Department of Theoretical Chemistry, Chemical Centre, P.O. Box 124, S-221 00 Lund, Sweden Received 8 November 1999; in final form 3 December 1999
Abstract The diagrammatic CASPT2 method is derived for an internally contracted basis. The wave-operator is efficient to compute and does not require orthogonalizing the internally contracted basis set and diagonalizing the zeroth-order Hamiltonian H0 . The wave-operator also has a simple form that is a generalization of Møller–Plesset perturbation theory. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The MRMP w1,2x and CASPT2 w3,4x methods are two ab initio methods that have been successful in computing electronic spectra of medium-sized molecules. However, these methods have their shortcomings. For example, the first-order wave-operator for MRMP has a different form for each determinantal state upon which it acts. This ket dependence makes the MRMP method less efficient, since the final energy is computed as a function of the transition density matrices involving the CASSCF state of interest < a : and each of the determinantal states within the CAS. The CASPT2 method avoids this shortcoming by using a non-orthogonal, internally contracted basis. Unfortunately, before computing the first-order wavefunction, CASPT2 requires steps to orthogonalize this non-orthogonal basis and diago-
nalize H0 , partially offsetting the advantage gained from using an internally contracted basis. The diagrammatic CASPT2 method w5x has a much more efficient formalism than either MRMP or CASPT2 but is designed to yields similar results. Recently the accuracy of diagrammatic CASPT2 has been demonstrated by computations of N2 , benzene and LiF w6x. Below we show that the first-order wave-operator
2. Diagrammatic CASPT2 theory
)
Fax: q81-3-5841-7241; e-mail: [email protected]
The first-order wave-operator V Ž1. is required to compute the second-order energy E Ž2.. This operator
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 1 4 4 7 - 5
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
generates the first-order wavefunction
V
Ž1. <
a : s
Ž 1.
where < a : is the CASSCF Žor CASCI. state of interest, < a : s Ý Cp < p: ,
Ž 2.
p
and < p : denotes the pth determinantal state from the CAS. The first-order wave-operator for MRMP, denoted Ž1. by V mp , is ket dependent, Ž1. Ž p. < : ² < V mp s Ý V mp p p.
The pth component of this wave-operator is given by w9x Ž p. V mp sÝ
wr
q 12
w r < h0 < w x † a a ´ˆ wr q D p r w w rw < sx x
Ý ´ˆ r s q D a†r a†s a x a w , p rswx w x
where the zeroth-order energies are given by Ei0 s ² i < H0 < i : .
Ž 4.
where
w r < h0 < w x † ar aw ´ˆ wr q Dˆwr
V dŽ1. s Ý wr
q 12
Ý rswx
´ˆ wr s eˆw y eˆr ,
Ž 5.
sx
´ˆ wr sx s eˆw q eˆ x y eˆr y eˆs ,
Ž 6.
q 12
Ž 7.
w ij < kl x are two-electron integrals written in chemist’s notation w10x; w and x denote inactive and active orbitals; while, r and s denote active and secondary orbitals, except that all indices cannot simultaneously be active Žno internal excitations.. Furthermore, the orbital energies eˆi define a diagonal, one-body, zeroth-order Hamiltonian,
Ž 8.
i Ž1. The ket dependence of V mp is caused by the shifts in the energy denominator D p that can can be interpreted as shifts that force the < p : states to be degenerate w9x,
D p s Ea0 y Ep0
,
Ž 9.
Ž 11 .
w s < h0 < x x s < aˆ x : ´ˆ xs q Dˆ xs
Ý rswx
a
H0 s Ý eˆi a†i a i .
w rw < sx x † † ar as a x aw , ´ˆ wr sx q Dˆwr sx
where the shifts, Dˆwr and Dˆwr sx , are at our disposal w11x, but are chosen in a manner so that the wave-operator V dŽ1. for diagrammatic CASPT2 is similar to the one used in the MRMP method, given by Eqs. Ž3. and Ž4.. Diagrammatic CASPT2 can also express its firstorder wave-function
i < h 0 < j s w i < h < j x q Ý Ž w ij < aa x y w ia < aj x . ,
Ž 10 .
The first-order wave-operator for diagrammatic Ž1. , CASPT2, denoted by V dŽ1., has the same for as V mp Ž . given by Eq. 4 , except that one- and two-body shifts, Dˆwr and Dˆwr sx , replace the ket dependent shift D p w5x,
Ž 3.
p
191
w rw < sx x < aˆ wr sx : , ´ˆ wr sx q Dˆwr sx
Ž 12 .
where the internally contracted states are generated by applying excitation operators to < a :, < aˆ xs : s a†s a x < a : ,
Ž 13 .
< aˆ wr sx : s a†r a†s a x a w < a : .
Ž 14 .
3. Energy denominators The one-body shift Dˆwr is given as a weighted average of the shifts D p for pth determinantal states < p : that can undergo w r excitations. Explicitly, they are given by
™
Dˆwr s
1 Nwr
Ý nŽwp. Ž 1 y nŽr p. . D p < C p < 2 , p
Ž 15 .
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
192
where
The following identities are proved in Appendix
Nwr s Ý nŽwp. Ž 1 y nŽr p. . < C p < 2 ;
Ž 16 .
p
C p2
the squared coefficient is the weight of < p : in the CASSCF state of interest, given by Eq. Ž2., and nŽi p. is the occupation – 0 or 1 – of the ith orbital in < p :. Similarly, the two-body shifts, that correspond to double excitations, are given by 1
ˆ
Dwr sx s
Nwr xs
Ý
nŽwp. nŽxp.
Ž
1 y nŽr p.
.Ž
1 y nŽs p.
. Dp < Cp <
2
1
Ý nŽwp. nŽmp. Ž 1 y nŽr p. . D p < C p < 2 , N rm
Ý
´ˆ wr sx q Dˆwr sx s Ea0 y E˜raw0s x ,
Ž 26 .
where E˜raw0 and E˜raw0s x are zeroth-order energies, E˜raw0 s
² aˆ wr < H0 < aˆ wr : ² aˆ wr < aˆ wr :
Ž
1 y nŽr p.
. < Cp < 2 ,
Ž 27 .
² aˆ wr sx < H0 < aˆ wr sx :
E˜raw0s x s
Ž 18 .
of the normalized, internally contracted states, given by Eqs. Ž13. and Ž14.. Using the identities, given by Eqs. Ž25. and Ž26., Eq. Ž12. can be written as
² aˆ wr sx < aˆ wr sx :
,
Ž 28 .
w s < h0 < x x s < aˆ x : Ea0 y E˜sax0
Ž 19 .
p
nŽwp. nŽmp.
,
Ž 17 .
The two-body shift correspond to single excitations whenever either w or x is an active orbital, say m, and either r or s is the same active orbital. These types of excitations are defined by
Nwr xs s
Ž 25 .
,
p
wm
´ˆ wr q Dˆwr s Ea0 y E˜raw0 ,
p
Nwr xs s Ý nŽwp. nŽxp. Ž 1 y nŽr p. .Ž 1 y nŽs p. . < C p < 2 .
Dˆwr mm s
A:
q 12
Ý rswx
Ž 20 .
w rw < sx x < aˆ wr sx : . Ea0 y E˜raw0s x
Ž 29 .
p
where all two-body shifts corresponding to single excitations are equal, mr rm Dˆmw s Dˆmw s Dˆwmmr s Dˆwr mm .
Ž 21 .
Eqs. Ž17. and Ž19. can be combined into a single relation that defines all two-body shifts,
Dˆwr sx s
1 Nwr xs =Ž
Ý
Using a restricted set of spin orbitals, the waveoperator for diagrammatic CASPT2 can be written as
V dŽ1. s Ý
Gr nŽwp. nŽxp. 1 y nŽr p. w x
Ž
4. Restricted formalism
.
wr
Ž r < h0 < w . ´ wr q Dwr
Eˆr w q
Ž rw < sx .
1 2
Ý rswx
´ wr sx q Dwr sx
Eˆr w s x ,
p
Ž 30 .
Gs 1 y nŽs p. w x D p < C p < 2
.
Nwr xs s Ý nŽwp. nŽxp. Ž 1 y nŽr p. .
G wr x
,
Ž 1 y nŽs p. .
G ws x
Ž 22 .
where
2
´ wr s e w y e r ,
Ž 31 .
´ wr sx s e w q e x y e r y e s ,
Ž 32 .
< Cp < ,
p
Ž 23 .
Ž i < h0 < j . s Ž i < h < j . q Ý 2 Ž ij < aa . y Ž ia < aj .
where
,
a
Ž 24 .
Ž 33 .
Note that terms involving Ž Gwix s y1. are not considered, since w s x terms do not occur in the first-order wave-operator, given by Eq. Ž12..
and Ž ij < kl . are spin independent two-electron integrals written in chemist’s notation w10x. Henceforth, the indices w, x, r, and s refer to spatial orbitals;
Gwix s 1 y d i w y d i x .
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
spin orbitals are denoted by ws , x s , rs , and ss , where s s a or b . The spin-adapted excitation operators w12x, or unitary group generators, given in Eqs. Ž34. and Ž35.,
s
Ý a†rs a†ss s
1
Ý N rs nŽwp.s Ž 1 y nŽrsp. . D p < C p < 2 ,
Dwr s 12 Ý Dwr sx s 14
X
X
a x s X a ws ,
Ž 35 .
1
Ý Ý N rs ss
ss X p
H0 s Ý e i Eˆi i ,
Ž 36 .
= Ž 1 y nŽs sp.X .
G ws ss x s X
Dp < Cp < 2 ,
where e i s e i s ; therefore, the following identities are valid and independent of s and s X : ,
Ž 46 .
p Ž p. Nwrss xs ss X s Ý nŽwp.s nŽxp. s X Ž1 y n rs .
X
Ž 38 .
By substituting Ž30. into Eq. Ž1. we get an expression for the first-order wave-function,
q 12
Ž s < h0 < x . ´ xs q D xs
< a xs :
Ž rw < sx .
Ý
´ wr sx q Dwr sx
rswx
< a wr sx : ,
Ž 39 .
X
= Ž 1 y nŽs sp.X .
ˆ
< a xs : s Es x < a : ,
Ž 40 .
< a wr sx : s Eˆr w s x < a : .
Ž 41 .
A reasonable, spin-adapted, modification of Eqs. Ž15. and Ž22. is given by an average over all possible spin combinations,
Dwr s 12 Ý Dˆwr ss , X
,
Ž 43 .
are given by Eqs. Ž15. and Ž22., with r, w, s, x,
Ž 48 .
X
Ž 49 .
By adding together the equations involving all spin states and using Eqs. Ž27., Ž28., Ž37., Ž38., Ž42. and Ž43. we get the following identities:
´ wr q Dwr s Ea0 y Eraw0 ,
Ž 50 .
´ wr sx q Dwr sx s Ea0 y Eraw0s x ,
Ž 51 .
where Eraw0 and Eraw0s x are average zeroth-order energies, Esax0 s 12 Ý
² aˆ xsss < H0 < aˆ xsss :
s
ss X
X where the spin-dependent shifts, Dˆwr ss and Dˆwr ss sxss X ,
Ž 47 .
´ˆ wr ss sxss X q Dˆwr ss sxss X s Ea0 y E˜ras0ws s s X x s X .
Ž 42 . X
< Cp < 2 ,
´ˆ wr ss q Dˆwr ss s Ea0 y E˜ras0ws ,
s
Ý Dˆwrss sxss
G ws ss x s X
and nŽi sp. is the occupation – 0 or 1 – of the ith orbital in < p : with s spin. Eqs. Ž44. and Ž45. differ slightly from the previous formulation of diagrammatic CASPT2 w5,6x. ŽNote that the equations for the energy denominator shifts in Ref. w5x have errors that are pointed out in Ref. w6x.. By substituting rs , ws , ss X , x s X for r, w, s, x, respectively, Eqs. Ž25. and Ž26. become
X
where the internally contracted states are generated by applying the excitation operator to < a :,
G wrss x s X
p
Ž 37 .
´ wr sx s ´ˆ wr ss sxss X .
Ž 45 .
Nwrss s Ý nŽwp.s Ž 1 y nŽr sp. . < C p < 2 , X
ˆ
G wrss x s X
where the weights are
i
Dwr sx s 14
ws x s
Ž p. nŽwp.s nŽxp. s X Ž1 y n rs .
X X
X
insure that our formalism is spin-adapted. Furthermore, the orbital energies e i are eigenfunctions of a diagonal one-body zeroth-order Hamiltonian,
´ wr s ´ wr ss
Ž 44 .
ws
p
Ž 34 .
s
Eˆr w s x s Ý
replaced by rs , ws , ss X , x s X , respectively. Explicitly, these equations are given by
s
Eˆr w s Ý a†r s a ws ,
193
² aˆ xsss < aˆ xsss :
,
Ž 52 .
X
Eraw0s x s 14
Ý
ss X
X
² aˆ wr ss sxss X < H0 < aˆ wr ss sxss X : X
X
² aˆ wr ss sxss X < aˆ wr ss sxss X :
,
Ž 53 .
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
194
of the normalized, spin-dependent, internally contracted states, given by < aˆ xsss : s a†s s a x s < a : , X
< a wr ss sxss X : s a†r s
ˆ
Ž 54 .
a†s s X a x s X a ws
Ž 55 .
As an alternative to Eqs. Ž52. and Ž53., Esax0 and a0 Er w s x can be taken as a generalization of Eqs. Ž27. and Ž28., Esax0 s
² a xs < H0 < a xs : ² a xs < a xs :
Eraw0s x s
,
Ž 56 .
² a wr sx < H0 < a wr sx : ² a wr sx < a wr sx :
,
Ž 57 .
where the internally contracted states are given by Eqs. Ž40. and Ž41.. The corresponding one- and two-body shifts, Dwr and Dwr sx , can then be computed using Eqs. Ž50. and Ž51.. Using the identities Žor definitions., given by Eqs. Ž50. and Ž51., Eq. Ž39. can be written as
Ž s < h0 < x .
Ea0 y Esax0
sx
q 12
rswx
Ea0 y Eraw0s x
< a wr sx : .
Ž 58 .
When one or more active electrons are present, we can express the first-order wave-operator in a form with the single excitations combined with the double excitations,
5. Comparison with single-reference perturbation theory Our formalism reduces to second-order Møller– Plesset perturbation theory when there are no active orbitals. For a single-determinantal reference-state, denoted by < a:, the first-order wave-operator is given by Eq. Ž1. with < a: replacing < a :,
V Ž1. < a: s
Ý Cwr sx < a˜ wr sx : ,
Ž 59 .
Ž 62 .
where the first-order wave-operator,
q 12
w s < h0 < x x s < aˆ x : Ea0 y E˜sa0x
Ý rswx
< a xs :
Ž rw < sx .
Ý
set without orthogonalizing these basis functions and diagonalizing H0 .
w rw < sx x r s < aˆ w x : , Ea0 y E˜ra0w s x
Ž 63 .
is identical to Eq. Ž29., except that the Žzeroth-order. excited states are generated using the single determinantal reference state, < aˆ sx : s a†s a x < a: ,
Ž 64 .
< aˆ wr sx : s a†r a†s a x a w < a: .
Ž 65 .
Furthermore, the zeroth-order energies are given by the expectation value of the zeroth-order Hamiltonian,
rwsx
where the internally contracted states < a˜ wr sx : and coefficients Cwr sx are given by
ˆ Eˆs x < a : ,
< a wr sx : s Er w
˜
Cwr sx s
Ž rw < sx . Ea0 y Eraw0s x yÝ m
q
Ž 60 . dr w
2 Ž s < h0 < x .
nÕ
Ea0 y Esax0
Ž sm < mx . a0 Ea0 y Esm mx
,
Ž 61 .
and the summation over m includes only active orbitals. Hence, unlike CASPT2, the first-order wave-operator
E˜sa0x s ² aˆ sx < H0 < aˆ sx : ,
Ž 66 .
E˜ra0w s x s ² aˆ wr sx < H0 < aˆ wr sx : ,
Ž 67 .
and Ea0 is given by Eq. Ž10.. Eqs. Ž27. and Ž28. are generalizations of Eqs. Ž66. and Ž67.. The additional denominator factors for Eqs. Ž66. and Ž67. seem appropriate, since the internally contracted states, < aˆ xs : and < aˆ wr sx :, are not normalized. For a one-body zeroth-order Hamiltonian, the energy denominators can be written as Ea0 y E˜sa0x s ´ˆ wr ,
Ž 68 .
Ea0 y E˜ra0w s x s ´ˆ wr sx ,
Ž 69 .
J.P. Finleyr Chemical Physics Letters 318 (2000) 190–195
where ´ˆ wr and ´ˆ wr sx are given by Eqs. Ž5. and Ž6.. These equations are identical to Ž25. and Ž26., except for the absence of the one- and two-body shifts, Dˆwr and Dˆwr sx . These shifts only depend on active indices, and vanish when no orbitals are active. Because of Brillouin’s theorem, the term corresponding to single excitations in Eq. Ž63. vanishes. Similarly, for a CASSCF reference state < a :, single excitations vanishes when x is inactive and s is secondary w13x.
the numerator factors in Eqs. Ž27. and Ž28. are also easily evaluated, ² aˆ wr < H0 < aˆ wr : s Ž eˆr y eˆw . ² aˆ wr < aˆ wr : q Ý nŽwp. Ž 1 y nŽr p. . < C p < 2 Ep0 , p
Ž A.5 . ² aˆ wr sx < H0 < aˆ wr sx : s Ý nŽwp. nŽxp. Ž 1 y nŽr p. .
G wr x
p
= Ž 1 y nŽs p. .
Acknowledgements
˚ Malmqvist The author would like to thank P.-A. for useful discussions.
195
G ws x
< C p < 2 Ep0
y ´ˆ wr sx² aˆ wr sx < aˆ wr sx : .
Ž A.6 .
Appendix A. Proofs for Eqs. (25) and (26) identities
By substituting Eqs. Ž15. and Ž27. into Ž25., and using ŽA.1., ŽA.5., Ž5., and Ž9., the Eq. Ž25. identity is proved. By substituting Eqs. Ž22. and Ž28. into Ž26., and using ŽA.2., ŽA.6., Ž6., and Ž9., the Eq. Ž25. identity is proved.
Using Eqs. Ž13. and Ž14., the denominators in Eqs. Ž27. and Ž28. are easily evaluated, yielding
References
² a wr < a wr : s Nwr
ˆ ˆ
,
Ž A.1 .
² aˆ wr sx < aˆ wr sx : s Nwr xs ,
Ž A.2 .
where Nwr and Nwr xs are given by Eqs. Ž16. and Ž23.. Since internal excitations are not considered, Ž w / r . for Eq. Ž25.. Also, Ž w / x . and Ž r / s . cases are not considered for Eq. Ž26., since these terms are zero for the first-order wave-operator: they violate the Pauli exclusion principle. Using the identities given by a†r a w H0ˆ < p : s a†r a w < p : Ž eˆr y eˆw q Ep0 . ,
Ž A.3 .
a†r a†s a x a w H0ˆ < p : s a†r a†s a x a w < p : eˆr q eˆs y eˆw y eˆ x q Ep0 ,
ž
/
Ž A.4 .
w1x K. Hirao, Chem. Phys. Lett. 190 Ž1992. 374. w2x K. Hirao, Chem. Phys. Lett. 196 Ž1992. 397. ˚ Malmqvist, B.O. Roos, A.J. Sadlej, K. w3x K. Andersson, P.-A. Wolinski, J. Phys. Chem. 94 Ž1990. 5483. ˚ Malmqvist, B.O. Roos, J. Phys. Chem. w4x K. Andersson, P.-A. 96 Ž1992. 1218. w5x J.P. Finley, J. Chem. Phys. 108 Ž1998. 1081. w6x J.P. Finley, H.A. Witek, J. Chem. Phys. in press. w7x R.J. Bartlett, D.M. Silver, J. Chem. Phys. 62 Ž1975. 325. w8x J.A. Pople, J.S. Binkley, R. Seeger, Int. J. Quantum Chem. 10S Ž1976. 1. w9x J.P. Finley, Chem. Phys. Lett. 283 Ž1998. 277. w10x A. Szabo, N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, 1st edn., Macmillian, New York, 1982. w11x J.P. Finley, J. Chem. Phys. 109 Ž1998. 7725. w12x B.O. Roos, P. Linse, P.E.M. Siegbahn, Chem. Phys. 66 Ž1981. 197. w13x B. Levy, G. Berthier, Int. J. Quantum Chem. 2 Ž1968. 307.