- Email: [email protected]
A self-consistent quasidegenerate coupled-cluster theory
1 October 1999
Chemical Physics Letters 311 Ž1999. 372–378 www.elsevier.nlrlocatercplett
A self-consistent quasidegenerate coupled-cluster theory Mark R. Hoffmann ) , Yuriy G. Khait
1
Department of Chemistry, UniÕersity of North Dakota, Grand Forks, ND 58202-9024, USA Received 14 June 1999
Abstract A new multireference coupled-cluster theory is suggested which extends the idea of a self-consistent primary space introduced earlier ŽJ. Chem. Phys. 108 Ž1998. 8317. in the context of quasidegenerate perturbation theory. The suggested approach, which we refer to as the self-consistent quasidegenerate coupled-cluster ŽSC-QDCC. method, is developed within a Hilbert space framework. The method is both stable to intruder states and reduces to a correct single-reference coupled-cluster theory; moreover, it is appropriate for both strongly and weakly quasidegenerate systems. The theory uses canonical normalization and produces hermitian effective Hamiltonians. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Although a number of multireference coupledcluster ŽMRCC. theories ŽFock space CC w1–7x, Hilbert space CC w8–14x. have been investigated, the question of a theory appropriate for strongly, as well as weakly, quasidegenerate systems cannot be considered solved. In this Letter, we present a new MRCC approach, within an n-body Hilbert space framework, which permits one to describe correctly eigenvalue spectra of varied quasidegeneracy and which avoids the intruder state problem w11,15–17x in a mathematically rigorous way. The proposed approach is a true CC formalism that includes nonlinear cluster amplitudes although linearization is possible and may be potentially interesting. We re-
) Corresponding author. Tel.: q1-701-777-2742; fax: q1-701777-2331; e-mail: [email protected] 1 Permanent address: 14 Dobrolyubova Avenue, Russian Scientific Center ‘Applied Chemistry’, St. Petersburg 197198, Russia.
strict our consideration herein to a general formalism of the method and to a discussion of its basic equations. The approach is based on three complementary ideas. First, the concept of a correct Žoptimal. primary subspace within an extended model space Ž L M ., which provides a zero-order approximation to both the lowest states of interest and to the low-lying excited Žsecondary. states, is used to restrict the domain of the wave operator that generates the exact wavefunctions. Second, we suggest a more general form of the wave operator which includes the wellestablished formalism of Jeziorski and Monkhorst w8x as a particular case. Third, invariances in the representation of the optimal primary subspace are exploited to give computationally appealing expressions for calculation of the correlated wavefunctions. Although not essential, the suggested approach uses unitary wave operators and forms hermitian effective Hamiltonians. As in the case of our previously presented quasidegenerate perturbation theory ŽSC-QDPT. w18x,
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 8 8 3 - 0
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
it will be shown that it is most reasonable to use the projection of the target wavefunctions on L M as the reference primary subspace. With this construction, the cluster operator is only required to describe interactions between the correct primary functions and high-lying Žexternal. configurations outside of L M . The correct primary functions must be determined self-consistently by a partial diagonalization of the effective Hamiltonian within L M at each iteration. Hence, the interactions between the primary and secondary states are described correctly Ži.e., untruncated variationally., even when these states are strongly quasidegenerate, and the problem of overlapping spectra of the secondary and external states is circumvented. We refer to this approach as the self-consistent quasidegenerate CC ŽSC-QDCC. method in analogy with the SC-QDPT method w18x, which also optimizes a primary subspace but describes the primary-external couplings perturbatively.
™
tions from the set
™
Ž 2.
Eqs. Ž1. and Ž2. lead to the generalized Bloch equation for determining the wave operator HUP s UPH eff P , ™
Ž 3. ™
where P s
Let us assume that we ™ are interested in determining an orthonormal set
™
H
Ž 1.
where E is a diagonal matrix of the states’ energies. Let an extended model space L M , which provides a zero-order approximation to both the primary and secondary states, be given. For example, the union of CASSCF wavefunctions and their complements, or the constituent configuration state functions, may™be used as a basis for L M . Then the projection of
Ž 4.
Using the optimal primary subspace in Eq. Ž2. significantly simplifies the structure of the wave operator ™ to be constructed. Indeed, since the projection of
2. Wave operator and effective Hamiltonian
373
Ž 5.
where S is the projector on LS . Eqs. Ž3. and Ž5. only determine the QUP and SUP blocks of the wave operator Žwhere Q is the orthogonal projector on the external space L Q of excited configurations. but not the PUP and Ž S q Q .UŽ S q Q . blocks; these blocks are arbitrary. Consequently, Eq. Ž3. and Ž5. are invariant relative to the choice of a basis within L P ; and, furthermore, in the case of using an exponential representation for U U s e X Ž Xqs yX . ,
Ž 6.
one can select the conceptually simple choice of PXP s 0 ,
XS s 0 .
Ž 7.
Such choice insures that Eq. Ž5. is satisfied and leads to the simple commutative relationships US s SU s S .
Ž 8.
Most significantly, Eq. Ž7. show that it is sufficient to take into account only the primary-external Ž P–Q . interactions in the wave operator. Primary-secondary Ž P–S . and secondary-external Ž S–Q . couplings may be omitted. In describing the P–Q interactions using an unitary wave operator and defining separate excitation, T, and de-excitation operators, Tq, X s T y Tq ,
Ž 9.
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
374
Eq. Ž7. restrict the form of T, without approximation, to T s QTQ q QTP .
Ž 10 .
Operators of such form are easily seen to satisfy the equations TqPM s 0 ,
PM TPM s 0 ,
Ž 11 .
where PM s P q S is the projector on L M . Eq. Ž11. permit us to restrict our considerations to an excitation operator that only produces excitations from L P external to the entire model space. Now let us consider construction of the optimal primary subspace L P within the given model space. Eq. Ž3. is equivalent to the pair of decoupling conditions SH eff P s 0 ,
Ž 12 .
QH eff P s 0 ,
Ž 13 .
which require that the subspace L P is stable relative to the effective Hamiltonian which only describes the P–Q interactions. Based on Eq. Ž12., such subspace can be constructed by partial diagonalization of the effective Hamiltonian matrix within L M for its NP-lowest eigenvectors. However, the excitation operator Žand, hence, H eff . itself depends on L P . Hence, Eq. Ž12. and Ž13. can only be solved iteratively; and the optimal primary subspace and the excitation operator T must correspond to their selfconsistent solution. The conditions for the convergence of the iterative construction of the wave operator and the subspace L P and a method for ensuring the convergence of the iterative process have been discussed earlier w18x. It has been shown, in particular, that the initial approximation to the optimal primary subspace proves not to be critical for the convergence. In addition, if the initial primary subspace is close to the optimal one, no iterative process for its improvement may be required. As seen from the relationships eff HSS s HSS ,
Ž 14 .
HSeffP s HS P UP P q HSQ UQ P ,
Ž 15 .
™
™
HPeffP s ²F P < Uq Ž P q Q . H Ž P q Q . U
Ž 16 .
which relate the H and H eff matrices in the model space, the P–S and S–Q interactions are also taken into account in the SC-QDCC method but, in contrast to the P–Q couplings, these interactions are described through the construction and partial diagonalization of H eff within L M . This treatment permits one to avoid completely the problem of possible quasidegeneracy between secondary and external states and, also, to describe the interactions between the primary and secondary states correctly even when these states are strongly quasidegenerate.
3. Unitary coupled-cluster ansatz The wavefunctions
Ž 17 .
for constructing the cluster and wave operators. Using this basis, the target wavefunctions from Eq. Ž2. can be written as
Ž 18 .
b
where P s < ™ w P :² ™ w P < s Ý a Pa and Pa s < wa :² wa < is the orthogonal projector on the given many-particle basis function wa . Let us define the action of the excitation operator T on the given function wa as T < wa : s Ý t Ia e I < wa : ,
Ž 19 .
I
where t Ia are cluster amplitudes to be determined; e I 4I is a common set of suitable individual manybody excitation operators, which, due to Eq. Ž11., produce only external excitations relative to L M , PM e I PM s 0 ,
eq I PM s 0 Ž ;I . ;
Ž 20 .
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
and I is a collective label for indices of the excitation operators. Thus, with every basis function wa we associate a distinct family t Ia4I of independent cluster amplitudes and a distinct cluster operator
375
and the one suggested by Jeziorski and Monkhorst w8x a
UJM P s Ý e T Pa
Ž 28 .
a
T a s Ý t Ia e I .
Ž 21 .
can be seen from a comparison of their expansions
I
As a consequence of Eq. Ž20., the cluster operators T a describing the P–Q interactions have the structure a
a
a
T s QT Q q QT Pa .
UP s P q Ý T a Pa a
1 q 2
Ž 22 .
q
Ý Ý Ž T b y T b . T aPa q . . . a
Ž 29.
b
1
Defining the total cluster operator T from Eq. Ž10. as a superposition
UJM P s P q Ý T a Pa q
Ts Ý T a ,
In the case of using the wave operator of Jeziorski and Monkhorst, one assumes, in fact, that cluster functions excited from different primary functions are orthogonal; in contrast, our form of the wave operator takes into account the possible non-orthogonality of these excited functions. In the simplest case of a one-dimensional primary subspace, the wave operators are identical Žexcept, of course, for normalization.. Hence, one may view the P–Q part of the present approach as a generalization of the Jeziorski and Monkhorst formalism.
Ž 23 .
a
one gets T s QTQ q Ý QT a Pa ,
Ž 24 .
a
where QTQ s Q
T a QsQ
žÝ / a
žÝ /
tI eI Q
Ž 25 .
I
and t I 4I is a set of state-averaged cluster amplitudes, t I s Ý t Ia Ž ;I . .
a
2
Ý Ž T a . 2 Pa q . . .
Ž 30.
a
Ž 26 .
a
4. Cluster amplitudes equations
The main feature of a total cluster operator T, defined thus, is that this operator acts on each basis function wa through the individual cluster operator T a but acts on any excited function from LQ through an ‘averaged’ cluster operator. This construction of T preserves maximum flexibility in the first application of the excitation operator without computationally prohibitive requirements due to the treatment of excited configurations. It is useful at this point to compare briefly the formalism suggested in the present work with the one of Jeziorski and Monkhorst, which has served as a basis for many state-universal w9–14x and statespecific MRCC approaches Žsee Refs. w19,20x and references therein.. The difference between the wave operator suggested in the present work
Ý ŽT UP s Ý e a
b
b
yT
Let us consider the case that the set e I 4I of direct individual excitations operators may be divided into subsets, e IŽkk .4Ik, of k-body excitation operators Ž k s 1,2, . . . . from the model space. Such splitting induces the decomposition of the external space and its projector, LQ s LQ 1 [ LQ 2 [ PPP
Ž 31 .
Q s Q1 q Q 2 q PPP ,
Ž 32 .
where Q k is the orthogonal projector on the subspace L Q k spanned by k-body excited configurations from L M . This also leads to expansions of cluster operators Ts
Ý Tk ,
Ž 33 .
kG1
b q.
Pa
Ž 27 .
T as
Ý Tka , kG1
Ž 34 .
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
376
where, in analogy with Eq. Ž21., the k-body cluster operators Tka, defined as Tka s Ý t ka, Ik e IŽkk . ,
Ž 35 .
Ik
are determined by state-specific k-body cluster amplitudes t k,a Ik. By definition, one has Tka Pa s Q k Tka P ,
Tk Q1 s Q1qk Tk Q1 ;
Ž 36 .
and, hence, the total cluster operator from Eq. Ž10. can be written in the form Ts
½
Ý Ý Q1q k Tk Q1 q Ý Qk TkaPa kG1
a
lG1
5
,
Ž 37 .
žÝ
/
Ž 38 .
and determined by state-averaged k-body cluster amplitudes t k , Ik s Ý t ka, Ik .
Ž 39 .
a
Due to the decomposition shown in Eq. Ž30., the Hamiltonian operator has a penta-diagonal form within the L P [ L Q space. This suggests the division of the Hamiltonian H s HD q HX
Ž 40 .
into its diagonal Ž H D . and off-diagonal Ž H X . parts, H D q PHP q
Ý Q k HQk ,
Ž 41 .
kG1
q Q k F Ž T . Pa Ž a s 1,2, . . . NP ; k s 1,2, . . . . ; Ž 45 . where the operator F ŽT . is defined as
Ž 46 .
kG1
qQ k H Ž Q kq1 q Q kq2 . .
Taking into account Eq. Ž32. and Ž40., the generalized Bloch equation, Eq. Ž3., may be written as a system of equations Q k w H D ,U x P s yQ k H X UP q Q k UP Ž H eff y H D . P ,
Ž 43 .
1
™
™
™
s yh ak q f ka Ž a s 1,2, . . . , NP ; k s 1,2, . . . . .
Ž 48 . ™a t k is the vector w™ t ka x Ik s t k,a Ik; Hkaka
In Eq. Ž48., of amplitudes with components and Skaka are the Hamiltonian and overlap matrices in the basis of the cluster functions e IŽkk . < wa :,
and
Ž 42 .
2
X 2q
Ž Hkaka y ´ a Skaka . t ka
Skaka
Ý Ž Q kq 1 q Qkq2 . HQk
1
X 3q... . Ž 47. 3! Now, projecting Eq. Ž45. from the right against < wa : and from the left against the cluster functions e IŽkk . < wa : and taking Eq. Ž17. into account, one obtains the desired set of matrix equations for all the state-specific cluster amplitudes t k,a Ik, U2 s U y X y 1 s
Hkaka
H X s Ž Q1 q Q2 . HP q PH Ž Q1 q Q2 . q
Q k H D ,Tka Pas y Q k HPa
and
t k , Ik e ŽIkk . Q1
Ik
Ž 44 .
where Pa s < wa :² wa <. Based on Eqs. Ž43. and Ž44., one obtains a system of non-linear equations determining the state specific k-body cluster operators Tka
q Q w U2 , H D x P
Tka Q1
žÝ / a
s Q kq 1
Q k w H D , X x Pa s Q k H D ,Tka Pa ,
F Ž T . s yQHX Ž U y 1 . P q QUP Ž H eff y H D . P
In analogy with Eq. Ž25., the blocks Q kq 1Tk Q1 s Q kq1
which is easily shown to be equivalent to Eq. Ž13.. In the basis of the eigenfunctions wa4a of the Hamiltonian matrix in a giÕen L P Žsee Eq. Ž17.. one also has
™a hk
Žk. Žk. I k , J k s ² e I k wa < H < e J k wa : , Žk. Žk. I k , J k s ² e I k wa < e J k wa : ,
™a hk
Ik
and
™ f ka
Ž 50 .
are vectors with components
s ² e IŽkk .wa < H < wa : ,
™ f ka I s ² e IŽkk . a < F k
w
Ž 49 .
Ž T . < wa : .
Ž 51 . Ž 52 .
In the case when the decomposition in Eq. Ž31. is correct, the system of Eq. Ž48. is equivalent to Eq. Ž13. and, hence, is exact. Although this system can
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
only be solved iteratively, it is reasonable to expect rapid convergence under usual physical and mathematical circumstances. Specifically, diagonal elements of the matrix Ž Hkaka – ´ a Skaka . are expected to be comparable to differences in eigenvalues of primary and external states, which are energetically well separated by the secondary states. In such cases, the cluster amplitudes may be expected to be small; moreover, the first-order approximation T Ž1. to the exact solution, as determined from the simplest form of Eq. Ž45., Q k H D ,Tka Pa s yQ k HPa .
Ž 53 .
may also be expected to be quite good. The non-linearity introduced in Eq. Ž45. Žor, equivalently, in Eq. Ž48.. by the operator QF ŽT Ž1.. P will also be small in this case, because the first non-zero term from this operator expansion is equal to yQHX QT Ž1. P. This term will be of second order in T Ž1., since QHX Q ought to be of the same magnitude as QHX P and QHX P can be seen Že.g., from Eq. Ž53.. to be of first order in T Ž1.. Thus, at least in cases in which the primary and external states are energetically well separated by the secondary ones, the system of Eq. Ž48. will yield a single, unique solution for the cluster amplitudes t k,a Ik. Moreover, taking into account that Q k HP s 0 if k s 2, Eq. Ž53. implies that, in the framework of the SC-QDCC method, 3-body, 4-body Žand higher. cluster amplitudes will be negligibly small in comparison with 1- and 2-body amplitudes. Hence, one can expect that the T s T1 q T2 truncation and the truncation of F ŽT . to the first few terms in the SC-QDCC approach will be sufficiently accurate and reliable for describing even strongly correlated electronic states. This conclusion is corroborated by the highly encouraging results obtained earlier with using the SC-QDPTŽSD. method w18x, which is based, in fact, on Eq. Ž53. with k s 1 and k s 2 only.
5. Summary The SC-QDCC method suggested in the present work is aimed at providing a physically realistic and mathematically robust description of a few low-lying quasidegenerate states within a MRCC approach. In the cases in which the primary and external states are
377
energetically well separated by the secondary ones and the primary and secondary states are not exactly degenerate, highly-accurate wavefunctions may be expected with inclusion of few non-linear terms in the cluster amplitudes Žsee Eq. Ž46... The method is potentially exact since it gives the exact lowest eigenvalues and eigenfunctions of the Hamiltonian in the configuration space L M [ LQ if all clusters would be taken into account. Moreover, based on the analysis of Eq. Ž48. and in conjunction with numerical evidence from the SC-QDPT method w18x, one should expect that a T s T1 q T2 operator truncation should be quite reliable and sufficiently accurate for describing most quasidegenerate states. It may be recognized that linear truncation of Eq. Ž48., together with a T s T1 q T2 operator manifold truncation, will yield calculations dominated by matrix vector products of the same dimensionality as MR-CISD equations. Furthermore, inclusion of the quadratic contribution from F ŽT . will involve some ‘double-quadruple’ matrix elements similar to those encountered in conventional single-reference CCSD theories. Hence, we do not expect unusual resource demands Že.g., CPU time, disk storage. in computational implementations, and, indeed, efforts to develop efficient computer program are currently in progress.
Acknowledgements The authors gratefully acknowledge the US Office of Naval Research ŽGrant No. N00014-96-1-1049. for financial support of the research presented here.
References w1x S. Pal, C.M.L. Rittby, R.J. Bartlett, D. Sinha, D. Mukherjee, J. Chem. Phys. 88 Ž1988. 4357. w2x B. Jeziorski, J. Paldus, J. Chem. Phys. 90 Ž1989. 2714. w3x D. Mukherjee, S. Pal, Adv. Quantum Chem. 20 Ž1989. 292. w4x U. Kaldor, Theor. Chim. Acta 80 Ž1991. 427. w5x D. Mukhopadhyay, S. Mukhopadhyay, R. Chaudhuri, D. Mukherjee, Theor. Chim. Acta 80 Ž1991. 441. w6x C.M.L. Rittby, R.J. Bartlett, Theor. Chim. Acta 80 Ž1991. 469. w7x J.F. Stanton, R.J. Bartlett, C.M.L. Rittby, J. Chem. Phys. 97 Ž1992. 5560.
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M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
w8x B. Jeziorski, H.J. Monkhorst, Phys. Rev. A 24 Ž1981. 1668. w9x W.D. Laidig, P. Saxe, R.J. Bartlett, J. Chem. Phys. 86 Ž1987. 887. w10x B. Jeziorski, J. Paldus, J. Chem. Phys. 88 Ž1988. 5673. w11x J. Paldus, L. Pylypow, B. Jeziorski, in: U. Kaldor ŽEd.., Many-Body Methods in Quantum Chemistry, Lecture Notes in Chemistry, vol. 52, Springer, Heidelberg, 1989, p. 151. w12x L. Meissner, K. Jankowski, J. Wasilewski, Int. J. Quantum Chem. 34 Ž1988. 535. w13x L. Meissner, S.A. Kucharski, R.J. Bartlett, J. Chem. Phys. 91 Ž1989. 6187. w14x L. Meissner, R.J. Bartlett, J. Chem. Phys. 92 Ž1990. 561.
w15x J. Paldus, in: G.L. Malli ŽEd.., Relativistic and Electron Correlation Effects in Molecules and Solids, NATO ASI Series B: Physics, vol. 318, Plenum, New York, 1994, p. 207. w16x Ph. Durand, J.P. Malrieu, in: K.P. Lawley ŽEd.., Advances in Chemical Physics, vol. 67, Wiley, New York, 1987, p. 321. w17x P. Piecuch, J. Paldus, Phys. Rev. A 49 Ž1994. 3479. w18x Yu.G. Khait, M.R. Hoffmann, J. Chem. Phys. 108 Ž1998. 8317. w19x X. Li, J. Paldus, J. Chem. Phys. 107 Ž1997. 6257. w20x J. Meller, J.P. Malrieu, R. Caballol, J. Chem. Phys. 104 Ž1996. 4068.
Chemical Physics Letters 311 Ž1999. 372–378 www.elsevier.nlrlocatercplett
A self-consistent quasidegenerate coupled-cluster theory Mark R. Hoffmann ) , Yuriy G. Khait
1
Department of Chemistry, UniÕersity of North Dakota, Grand Forks, ND 58202-9024, USA Received 14 June 1999
Abstract A new multireference coupled-cluster theory is suggested which extends the idea of a self-consistent primary space introduced earlier ŽJ. Chem. Phys. 108 Ž1998. 8317. in the context of quasidegenerate perturbation theory. The suggested approach, which we refer to as the self-consistent quasidegenerate coupled-cluster ŽSC-QDCC. method, is developed within a Hilbert space framework. The method is both stable to intruder states and reduces to a correct single-reference coupled-cluster theory; moreover, it is appropriate for both strongly and weakly quasidegenerate systems. The theory uses canonical normalization and produces hermitian effective Hamiltonians. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Although a number of multireference coupledcluster ŽMRCC. theories ŽFock space CC w1–7x, Hilbert space CC w8–14x. have been investigated, the question of a theory appropriate for strongly, as well as weakly, quasidegenerate systems cannot be considered solved. In this Letter, we present a new MRCC approach, within an n-body Hilbert space framework, which permits one to describe correctly eigenvalue spectra of varied quasidegeneracy and which avoids the intruder state problem w11,15–17x in a mathematically rigorous way. The proposed approach is a true CC formalism that includes nonlinear cluster amplitudes although linearization is possible and may be potentially interesting. We re-
) Corresponding author. Tel.: q1-701-777-2742; fax: q1-701777-2331; e-mail: [email protected] 1 Permanent address: 14 Dobrolyubova Avenue, Russian Scientific Center ‘Applied Chemistry’, St. Petersburg 197198, Russia.
strict our consideration herein to a general formalism of the method and to a discussion of its basic equations. The approach is based on three complementary ideas. First, the concept of a correct Žoptimal. primary subspace within an extended model space Ž L M ., which provides a zero-order approximation to both the lowest states of interest and to the low-lying excited Žsecondary. states, is used to restrict the domain of the wave operator that generates the exact wavefunctions. Second, we suggest a more general form of the wave operator which includes the wellestablished formalism of Jeziorski and Monkhorst w8x as a particular case. Third, invariances in the representation of the optimal primary subspace are exploited to give computationally appealing expressions for calculation of the correlated wavefunctions. Although not essential, the suggested approach uses unitary wave operators and forms hermitian effective Hamiltonians. As in the case of our previously presented quasidegenerate perturbation theory ŽSC-QDPT. w18x,
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 8 8 3 - 0
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
it will be shown that it is most reasonable to use the projection of the target wavefunctions on L M as the reference primary subspace. With this construction, the cluster operator is only required to describe interactions between the correct primary functions and high-lying Žexternal. configurations outside of L M . The correct primary functions must be determined self-consistently by a partial diagonalization of the effective Hamiltonian within L M at each iteration. Hence, the interactions between the primary and secondary states are described correctly Ži.e., untruncated variationally., even when these states are strongly quasidegenerate, and the problem of overlapping spectra of the secondary and external states is circumvented. We refer to this approach as the self-consistent quasidegenerate CC ŽSC-QDCC. method in analogy with the SC-QDPT method w18x, which also optimizes a primary subspace but describes the primary-external couplings perturbatively.
™
tions from the set
™
Ž 2.
Eqs. Ž1. and Ž2. lead to the generalized Bloch equation for determining the wave operator HUP s UPH eff P , ™
Ž 3. ™
where P s
Let us assume that we ™ are interested in determining an orthonormal set
™
H
Ž 1.
where E is a diagonal matrix of the states’ energies. Let an extended model space L M , which provides a zero-order approximation to both the primary and secondary states, be given. For example, the union of CASSCF wavefunctions and their complements, or the constituent configuration state functions, may™be used as a basis for L M . Then the projection of
Ž 4.
Using the optimal primary subspace in Eq. Ž2. significantly simplifies the structure of the wave operator ™ to be constructed. Indeed, since the projection of
2. Wave operator and effective Hamiltonian
373
Ž 5.
where S is the projector on LS . Eqs. Ž3. and Ž5. only determine the QUP and SUP blocks of the wave operator Žwhere Q is the orthogonal projector on the external space L Q of excited configurations. but not the PUP and Ž S q Q .UŽ S q Q . blocks; these blocks are arbitrary. Consequently, Eq. Ž3. and Ž5. are invariant relative to the choice of a basis within L P ; and, furthermore, in the case of using an exponential representation for U U s e X Ž Xqs yX . ,
Ž 6.
one can select the conceptually simple choice of PXP s 0 ,
XS s 0 .
Ž 7.
Such choice insures that Eq. Ž5. is satisfied and leads to the simple commutative relationships US s SU s S .
Ž 8.
Most significantly, Eq. Ž7. show that it is sufficient to take into account only the primary-external Ž P–Q . interactions in the wave operator. Primary-secondary Ž P–S . and secondary-external Ž S–Q . couplings may be omitted. In describing the P–Q interactions using an unitary wave operator and defining separate excitation, T, and de-excitation operators, Tq, X s T y Tq ,
Ž 9.
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
374
Eq. Ž7. restrict the form of T, without approximation, to T s QTQ q QTP .
Ž 10 .
Operators of such form are easily seen to satisfy the equations TqPM s 0 ,
PM TPM s 0 ,
Ž 11 .
where PM s P q S is the projector on L M . Eq. Ž11. permit us to restrict our considerations to an excitation operator that only produces excitations from L P external to the entire model space. Now let us consider construction of the optimal primary subspace L P within the given model space. Eq. Ž3. is equivalent to the pair of decoupling conditions SH eff P s 0 ,
Ž 12 .
QH eff P s 0 ,
Ž 13 .
which require that the subspace L P is stable relative to the effective Hamiltonian which only describes the P–Q interactions. Based on Eq. Ž12., such subspace can be constructed by partial diagonalization of the effective Hamiltonian matrix within L M for its NP-lowest eigenvectors. However, the excitation operator Žand, hence, H eff . itself depends on L P . Hence, Eq. Ž12. and Ž13. can only be solved iteratively; and the optimal primary subspace and the excitation operator T must correspond to their selfconsistent solution. The conditions for the convergence of the iterative construction of the wave operator and the subspace L P and a method for ensuring the convergence of the iterative process have been discussed earlier w18x. It has been shown, in particular, that the initial approximation to the optimal primary subspace proves not to be critical for the convergence. In addition, if the initial primary subspace is close to the optimal one, no iterative process for its improvement may be required. As seen from the relationships eff HSS s HSS ,
Ž 14 .
HSeffP s HS P UP P q HSQ UQ P ,
Ž 15 .
™
™
HPeffP s ²F P < Uq Ž P q Q . H Ž P q Q . U
Ž 16 .
which relate the H and H eff matrices in the model space, the P–S and S–Q interactions are also taken into account in the SC-QDCC method but, in contrast to the P–Q couplings, these interactions are described through the construction and partial diagonalization of H eff within L M . This treatment permits one to avoid completely the problem of possible quasidegeneracy between secondary and external states and, also, to describe the interactions between the primary and secondary states correctly even when these states are strongly quasidegenerate.
3. Unitary coupled-cluster ansatz The wavefunctions
Ž 17 .
for constructing the cluster and wave operators. Using this basis, the target wavefunctions from Eq. Ž2. can be written as
Ž 18 .
b
where P s < ™ w P :² ™ w P < s Ý a Pa and Pa s < wa :² wa < is the orthogonal projector on the given many-particle basis function wa . Let us define the action of the excitation operator T on the given function wa as T < wa : s Ý t Ia e I < wa : ,
Ž 19 .
I
where t Ia are cluster amplitudes to be determined; e I 4I is a common set of suitable individual manybody excitation operators, which, due to Eq. Ž11., produce only external excitations relative to L M , PM e I PM s 0 ,
eq I PM s 0 Ž ;I . ;
Ž 20 .
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
and I is a collective label for indices of the excitation operators. Thus, with every basis function wa we associate a distinct family t Ia4I of independent cluster amplitudes and a distinct cluster operator
375
and the one suggested by Jeziorski and Monkhorst w8x a
UJM P s Ý e T Pa
Ž 28 .
a
T a s Ý t Ia e I .
Ž 21 .
can be seen from a comparison of their expansions
I
As a consequence of Eq. Ž20., the cluster operators T a describing the P–Q interactions have the structure a
a
a
T s QT Q q QT Pa .
UP s P q Ý T a Pa a
1 q 2
Ž 22 .
q
Ý Ý Ž T b y T b . T aPa q . . . a
Ž 29.
b
1
Defining the total cluster operator T from Eq. Ž10. as a superposition
UJM P s P q Ý T a Pa q
Ts Ý T a ,
In the case of using the wave operator of Jeziorski and Monkhorst, one assumes, in fact, that cluster functions excited from different primary functions are orthogonal; in contrast, our form of the wave operator takes into account the possible non-orthogonality of these excited functions. In the simplest case of a one-dimensional primary subspace, the wave operators are identical Žexcept, of course, for normalization.. Hence, one may view the P–Q part of the present approach as a generalization of the Jeziorski and Monkhorst formalism.
Ž 23 .
a
one gets T s QTQ q Ý QT a Pa ,
Ž 24 .
a
where QTQ s Q
T a QsQ
žÝ / a
žÝ /
tI eI Q
Ž 25 .
I
and t I 4I is a set of state-averaged cluster amplitudes, t I s Ý t Ia Ž ;I . .
a
2
Ý Ž T a . 2 Pa q . . .
Ž 30.
a
Ž 26 .
a
4. Cluster amplitudes equations
The main feature of a total cluster operator T, defined thus, is that this operator acts on each basis function wa through the individual cluster operator T a but acts on any excited function from LQ through an ‘averaged’ cluster operator. This construction of T preserves maximum flexibility in the first application of the excitation operator without computationally prohibitive requirements due to the treatment of excited configurations. It is useful at this point to compare briefly the formalism suggested in the present work with the one of Jeziorski and Monkhorst, which has served as a basis for many state-universal w9–14x and statespecific MRCC approaches Žsee Refs. w19,20x and references therein.. The difference between the wave operator suggested in the present work
Ý ŽT UP s Ý e a
b
b
yT
Let us consider the case that the set e I 4I of direct individual excitations operators may be divided into subsets, e IŽkk .4Ik, of k-body excitation operators Ž k s 1,2, . . . . from the model space. Such splitting induces the decomposition of the external space and its projector, LQ s LQ 1 [ LQ 2 [ PPP
Ž 31 .
Q s Q1 q Q 2 q PPP ,
Ž 32 .
where Q k is the orthogonal projector on the subspace L Q k spanned by k-body excited configurations from L M . This also leads to expansions of cluster operators Ts
Ý Tk ,
Ž 33 .
kG1
b q.
Pa
Ž 27 .
T as
Ý Tka , kG1
Ž 34 .
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
376
where, in analogy with Eq. Ž21., the k-body cluster operators Tka, defined as Tka s Ý t ka, Ik e IŽkk . ,
Ž 35 .
Ik
are determined by state-specific k-body cluster amplitudes t k,a Ik. By definition, one has Tka Pa s Q k Tka P ,
Tk Q1 s Q1qk Tk Q1 ;
Ž 36 .
and, hence, the total cluster operator from Eq. Ž10. can be written in the form Ts
½
Ý Ý Q1q k Tk Q1 q Ý Qk TkaPa kG1
a
lG1
5
,
Ž 37 .
žÝ
/
Ž 38 .
and determined by state-averaged k-body cluster amplitudes t k , Ik s Ý t ka, Ik .
Ž 39 .
a
Due to the decomposition shown in Eq. Ž30., the Hamiltonian operator has a penta-diagonal form within the L P [ L Q space. This suggests the division of the Hamiltonian H s HD q HX
Ž 40 .
into its diagonal Ž H D . and off-diagonal Ž H X . parts, H D q PHP q
Ý Q k HQk ,
Ž 41 .
kG1
q Q k F Ž T . Pa Ž a s 1,2, . . . NP ; k s 1,2, . . . . ; Ž 45 . where the operator F ŽT . is defined as
Ž 46 .
kG1
qQ k H Ž Q kq1 q Q kq2 . .
Taking into account Eq. Ž32. and Ž40., the generalized Bloch equation, Eq. Ž3., may be written as a system of equations Q k w H D ,U x P s yQ k H X UP q Q k UP Ž H eff y H D . P ,
Ž 43 .
1
™
™
™
s yh ak q f ka Ž a s 1,2, . . . , NP ; k s 1,2, . . . . .
Ž 48 . ™a t k is the vector w™ t ka x Ik s t k,a Ik; Hkaka
In Eq. Ž48., of amplitudes with components and Skaka are the Hamiltonian and overlap matrices in the basis of the cluster functions e IŽkk . < wa :,
and
Ž 42 .
2
X 2q
Ž Hkaka y ´ a Skaka . t ka
Skaka
Ý Ž Q kq 1 q Qkq2 . HQk
1
X 3q... . Ž 47. 3! Now, projecting Eq. Ž45. from the right against < wa : and from the left against the cluster functions e IŽkk . < wa : and taking Eq. Ž17. into account, one obtains the desired set of matrix equations for all the state-specific cluster amplitudes t k,a Ik, U2 s U y X y 1 s
Hkaka
H X s Ž Q1 q Q2 . HP q PH Ž Q1 q Q2 . q
Q k H D ,Tka Pas y Q k HPa
and
t k , Ik e ŽIkk . Q1
Ik
Ž 44 .
where Pa s < wa :² wa <. Based on Eqs. Ž43. and Ž44., one obtains a system of non-linear equations determining the state specific k-body cluster operators Tka
q Q w U2 , H D x P
Tka Q1
žÝ / a
s Q kq 1
Q k w H D , X x Pa s Q k H D ,Tka Pa ,
F Ž T . s yQHX Ž U y 1 . P q QUP Ž H eff y H D . P
In analogy with Eq. Ž25., the blocks Q kq 1Tk Q1 s Q kq1
which is easily shown to be equivalent to Eq. Ž13.. In the basis of the eigenfunctions wa4a of the Hamiltonian matrix in a giÕen L P Žsee Eq. Ž17.. one also has
™a hk
Žk. Žk. I k , J k s ² e I k wa < H < e J k wa : , Žk. Žk. I k , J k s ² e I k wa < e J k wa : ,
™a hk
Ik
and
™ f ka
Ž 50 .
are vectors with components
s ² e IŽkk .wa < H < wa : ,
™ f ka I s ² e IŽkk . a < F k
w
Ž 49 .
Ž T . < wa : .
Ž 51 . Ž 52 .
In the case when the decomposition in Eq. Ž31. is correct, the system of Eq. Ž48. is equivalent to Eq. Ž13. and, hence, is exact. Although this system can
M.R. Hoffmann, Y.G. Khaitr Chemical Physics Letters 311 (1999) 372–378
only be solved iteratively, it is reasonable to expect rapid convergence under usual physical and mathematical circumstances. Specifically, diagonal elements of the matrix Ž Hkaka – ´ a Skaka . are expected to be comparable to differences in eigenvalues of primary and external states, which are energetically well separated by the secondary states. In such cases, the cluster amplitudes may be expected to be small; moreover, the first-order approximation T Ž1. to the exact solution, as determined from the simplest form of Eq. Ž45., Q k H D ,Tka Pa s yQ k HPa .
Ž 53 .
may also be expected to be quite good. The non-linearity introduced in Eq. Ž45. Žor, equivalently, in Eq. Ž48.. by the operator QF ŽT Ž1.. P will also be small in this case, because the first non-zero term from this operator expansion is equal to yQHX QT Ž1. P. This term will be of second order in T Ž1., since QHX Q ought to be of the same magnitude as QHX P and QHX P can be seen Že.g., from Eq. Ž53.. to be of first order in T Ž1.. Thus, at least in cases in which the primary and external states are energetically well separated by the secondary ones, the system of Eq. Ž48. will yield a single, unique solution for the cluster amplitudes t k,a Ik. Moreover, taking into account that Q k HP s 0 if k s 2, Eq. Ž53. implies that, in the framework of the SC-QDCC method, 3-body, 4-body Žand higher. cluster amplitudes will be negligibly small in comparison with 1- and 2-body amplitudes. Hence, one can expect that the T s T1 q T2 truncation and the truncation of F ŽT . to the first few terms in the SC-QDCC approach will be sufficiently accurate and reliable for describing even strongly correlated electronic states. This conclusion is corroborated by the highly encouraging results obtained earlier with using the SC-QDPTŽSD. method w18x, which is based, in fact, on Eq. Ž53. with k s 1 and k s 2 only.
5. Summary The SC-QDCC method suggested in the present work is aimed at providing a physically realistic and mathematically robust description of a few low-lying quasidegenerate states within a MRCC approach. In the cases in which the primary and external states are
377
energetically well separated by the secondary ones and the primary and secondary states are not exactly degenerate, highly-accurate wavefunctions may be expected with inclusion of few non-linear terms in the cluster amplitudes Žsee Eq. Ž46... The method is potentially exact since it gives the exact lowest eigenvalues and eigenfunctions of the Hamiltonian in the configuration space L M [ LQ if all clusters would be taken into account. Moreover, based on the analysis of Eq. Ž48. and in conjunction with numerical evidence from the SC-QDPT method w18x, one should expect that a T s T1 q T2 operator truncation should be quite reliable and sufficiently accurate for describing most quasidegenerate states. It may be recognized that linear truncation of Eq. Ž48., together with a T s T1 q T2 operator manifold truncation, will yield calculations dominated by matrix vector products of the same dimensionality as MR-CISD equations. Furthermore, inclusion of the quadratic contribution from F ŽT . will involve some ‘double-quadruple’ matrix elements similar to those encountered in conventional single-reference CCSD theories. Hence, we do not expect unusual resource demands Že.g., CPU time, disk storage. in computational implementations, and, indeed, efforts to develop efficient computer program are currently in progress.
Acknowledgements The authors gratefully acknowledge the US Office of Naval Research ŽGrant No. N00014-96-1-1049. for financial support of the research presented here.
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