J. Phys. Chem. 1982, 86, 2133-2140
figurations. The remaining Q space is divided into a secondary S space and the tail T space (Q= S .)'2 The CI techniques drop the SHT and THS couplings as well as the off-diagonal components of THT. This CI technique is a direct analogue of our retention of a limited set of orbitals in third order &e., retention of off-diagonal portions of SHS to lowest order) and the use of the full basis in second order (i.e., retention of diagonal portions of SHS and THF'l). The CI technique of extrapolating to include all of the diagonal parts of THT is unnecessary in QDMBPT as the analogous second-order diagrams involve simple summations over orbitals and are quick and efficient to implement. In addition, the analogue of the CI size consistency Davidson correctionsz2are automatically
+
(21)In second order, only the diagonal portions of SHoSand THoT are retained; the diagonal portions of SVS and TVT appear first in the numerator of the third-order expansion. (22)E. R. Davidson, 'The World of Quantum Chemistry", R. Daudel and B. Pullman, Ed., Reidel, Dordrecht, 1974;S. R. Langhoff and E. R. Davidson, Int. J. Quantum. Chem., 8, 61 (1974).
2133
incorporated within the perturbative framework. The close similarity between the QDMBPT and CI methods suggests that the simplifications found here for the third-order QDMBPT silicon calculations are representative of a general feature of the perturbation method. These conclusions have important implications for ordinary nondegenerate MBPT calculations where fourth-order contributions from triply excited-configurations provide significant contributions to the fourth-order energy but require an order of magnitude more computer time than other fourth-order terms. Based on our silicon calculation a similar truncation in the orbital space is anticipated to be valid for the accurate (and fast) evaluation of these troublesome diagrams.
Acknowledgment. This research is supported, in part, by NSF Grant CHE80-23456. (23)S.Bashkin and J. 0. Stoner, 'Atomic Energy Levels and Grotrian Diagrams", Vol 1, North Holland, Amsterdam, 1975.
Quasidegenerate Perturbation Theory Gabrlel Hose+ and Uzl Kaldor' Chemistry Department, Tel-Aviv University, 89978 Tel Aviv, Israel (Received: July 20, 1981)
A diagrammatic perturbation theory is presented which is applicable to nondegenerate, quasidegenerate, or degenerate zero-order function spaces. The model space, in which the effective Hamiltonian is diagonalized, is selected on a state-by-state basis, and the requirement of a complete model space (all possible distributions of the electrons in the open shells) is dropped. This feature is particularly important if there is more than one open shell, as is the case in most atomic and molecular excited states. The partitioning of the orbitals into two classes, holes and particles, instead of the usual three (core,valence, and particles), leads to a reduction in the number of diagrams and to a greater degree of diagram cancellation. Certain unlinked diagrams appear for incomplete model spaces; the expansion is, however, shown to be size consistent in all cases. An application to the c32,+ and C1$+ states of Hez near an avoided crossing is described. While the nondegenerate theory diverges in this case and other degenerate theories require a 25-configurationmodel space, good convergence is obtained by the present method with a three-configuration model space and fewer diagrams.
I. Introduction The method of diagrammatic many-body perturbation theory (MBPT), a powerful approach to the study of correlation effects in nuclei, atoms, and molecules, originates from the work of Brueckner,' Goldstone,2 and Hugenh~ltz.~ It was fmt applied to atomic systems by Kelly? Others5 followed, and molecular applications soon appeared, first by single-center expansions6and more recently by using basis set methods.'-'O With the improvement of computational techniques,ll MBPT has been shown to be potentially competitive with the widely used configuration interaction method.12 The Brueckner-Goldstone diagrammatic expansion is applicable when the unperturbed wave function is representable by a single determinant. This is the case for closed-shell systems and for certain types of open shells, such as the half-filled shell with maximum spin. In other cases, degenerate perturbation theory must be used. The Hilbert space is partitioned13 so that an effective HamilDepartment of Chemistry, University of Southern California, Los Angeles, CA 90007.
tonian (or reaction operator) is diagonalized inside a space of low dimensionality,called the model space. Interactions (1) K. A. Brueckner, Phys. Rev., 100,36 (1955). (2)J. Goldstone, Proc. R. SOC.London, Ser. A, 239,267 (1957). (3)N. M. Hugenholtz, Physica, 23,481 (1957). (4)H. P. Kelly, Phys. Rev., 131, 684 (1963);136,B896 (1964);Adv. Chem. Phys., 14, 129 (1969). (5)See, e.g., E. S. Chang, R. T. Pu, and T. P. Das, Phys. Rev., 174,1 (1968);N. C. Dutta, C. Matsubara, R. T. Pu, and T. P. Das, ibid., 177, 33 (1969). (6)H. P. Kelly, Phys. Rev. Lett., 26,679 (1971);T. Lee, N.C. Dutta, and T. P. Das, ibid., 25,204 (1970);T. Lee and T. P. Das, Phys. Rev. A, 6 _ 968 , _ -(19721 _ ~
(7)J. M.Schulman and D. N. Kaufman, J. Chem. Phys., 53, 477 (1970);57, 2328 (1972). (8)U. Kaldor, Phys. Rev. Lett., 31, 1338 (1973);J. Chem. Phys., 62, 4634 (1975);63,2199 (1975);P. S. Stern and U. Kaldor, ibid., 64,2002 (1976). (9)M.A. Robb, Chem. Phys. Lett., 20,274(1973);D . Hegarty and M. A. Robb, Mol. Phys., 37, 1455 (1979). (10)R.J. Bartlett and D. M. Silver, J.Chem. Phys., 62,3258(1975); S. Wilson and D. M. Silver, Int. J. Quantum Chem., 12,737(1977);S . Wilson, Mol. Phys., 35, 1 (1978). (11)R. J. Bartlett and D. M. Silver in "Quantum Science", J. L. Calais, 0. Goscinski, J. Linderberg, and Y. Ohm, Ed., Plenum, New York, 1976, p 393.
0022-3654/82/2086-2l33$01.25/00 1982 American Chemical Society
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outside the model space are included perturbatively. The first linked, energy-independent MBPT for degenerate or quasidegenerate systems was derived by Brandow.14 Brandow's method has been applied to molecular excited states8p9and to hyperfine interactions in excited atoms.15 Another interesting application is the effort of Freed and his co1laborators"j to define and calculate parameters for semiempirical theories. Brandow's method, as well as subsequent formulations,l' starts by partitioning the orbital space into hole, valence, and particle orbitals. The model space is then constructed from determinants corresponding to all possible combinations of the valence orbitals with the right symmetry (a complete model space). This is a very reasonable choice of the model space for systems with one open shell, e.g., open-shell ground states of nuclei, atoms, and molecules. Many excited states have, however, two or more open shells which may be well separated in energy. This happens even for low-lying excited states, such as those obtained upon single excitation from a closed shell. A complete model space, comprising all combinations of orbitals from all the open shells, would be of large dimension and involve a great computational effort. The energy range spanned by such a model space would be broad, and many of the functions would contribute little to the state under investigation. Moreover, functions outside the model space and interacting with it may have energies falling within this broad energy range, causing slow or no convergence of the perturbation series. Another situation where a complete model space is not the best choice may arise near an avoided crossing of potential surfaces. Orbitals occupied in some but not all the determinants describing these surfaces must be treated as open shells in Brandow's method, leading to large model spaces. A theory applicable to an arbitrary model space, complete or incomplete, degenerate, quasidegenerate, or nondegenerate, has been proposed recently.'* It enables us to construct the model space by selecting states rather than orbitals. The concept of valence shells as used by Brandow loses i b significance and is in fact abandoned. The orbital space is partitioned instead into holes and particles according to the occupancy of the ket state on which the effective Hamiltonian operates, yielding diagrams which are generalizations of the Goldstone diagrams for closed shells. This definition of the reference state is closer to the actual wave function than Brandow's choice of the core state (no valence orbitals occupied). It leads to a greater degree of diagram cancellation before the computational (12)S. Wilson and D. M. Silver, Phys. Rev. A , 14,1949 (1976);D.L. Freeman and M. Karplus, J.Chem. Phys., 64,4578(1976);R. J. Bartlett and I. Shavitt, Chem. Phys. Lett., 50,190(1977);M. Urban, V.Kello, and I. Hubac, ibid., 51, 170 (1977). (13)P.-0. Lowdin, J. Chem. Phys., 19, 1396 (1951);J.Math. Phys., 3,969 (1962). (14)B. H.Brandow, Reu. Mod. Phys., 39, 771 (1967);in "Effective Interactions and Operators in Nuclei", B. R. Barrett, Ed., Springer, Berlin, 1975,p 1; Adu. Quantum Chem., 10, 187 (1977). (15)S.Garpman, I. Lindgren, J. Lindgren, and J. Morrison, Phys. Rev. A , 11, 758 (1975);L. Holmgren, I. Lindgren, J. Morrison, and A. M. Martensson, 2.Phys. A, 276, 179 (1976);I. Lindgren, J. Lindgren, and A. M. Martensson, ibid., 279, 113 (1976). (16)S.Iwata and K. F. Freed, J.Chem. Phys., 61,1500(1974);Chem. Phys. Lett., 28,176 (1974);D.L. Yaeger, H. Sun, K. F. Freed, and M. F. Herman, ibid., 57,490 (1978);H. Sun, K. F. Freed, M. F. Herman, and D. L. Yaeger, J. Chem. Phys., 72,4158 (1980). (17)P.G. Sandars, Adu. Chem. Phys., 14,365(1969);G. Oberlechner, Owono-N'-Guema, and J. Richert, Nuou. Cim. B, 68,23 (1970);M. B. Johnson and M. Baranger, Ann. Phys. N.Y., 62,172(1971);T.T.S.Kuo, S. Y. Lee, and K. F. Ratcliff, Nucl. Phys. A , 176,65(1971);I . Lindgren, J.phys. E , 7,2441(1974);V. Kvasnicka, Czech. J. Phys. B, 25,371(1975). (18)G. Hose and U. Kaldor, J. Phys. B, 12,3827 (1979);Phys. Scr., 21, 357 (1980).
stage and to a significant reduction in the number of diagrams to be calculated. It also facilitates infinite-order summation of classes of diagrams by denominator shifts.lg The theoretical background of the general-model-space perturbation theory is presented in the next section. The structure and summation rules of the generalized Goldstone diagrams are explained in section 111, and an application to E,+ excited states of Hez near a curve crossing is described in section IV. The final section includes a summary and conclusions. 11. Theory A. The Reaction Operator. Consider the Schrodinger equation
H\E = E* The Hamiltonian is partitioned into
(1)
H=Ho+V with the solutions of Ho known Ho@i= Ei9i
(2) (3)
A model space D is defined by selecting a manifold of zero-order states ai (model states). The state projection operator is
Pi =
(4)
I@i) (ail
and the projection operators onto and outside the model space are p = xpi Q = I - P = CPi (5) iED ieD The energy correction AE for an exactly degenerate model space is given by
AE=E-E,
(6)
where E, is the zero-order energy of the model states
H J ' = E$
(7)
In the case of a quasidegenerate model space, Ho and V are shifted14 by the diagonal perturbation V l
Ho' = Ho
+ v,
V' =
v - V'
H,' is exactly degenerate H,'P = E$
(9)
and the theory proceeds as in the degenerate case. V 1 obviously vanishes for an exactly degenerate model space. The usual Q-space Green's functions
are related through the Dyson equation G = Go + GoVG
(11)
The reaction operat~r,'~ also known as the effective Hamiltonian or level-shift operator,20is given by W = PVP + PVGVP (12) and satisfies the eigenvalue equation
w m = (AJ?3+ v')m
(13)
(19)G.Hose and U. Kaldor, Chem. Phys., 62,469 (1981). (20)H.Feshbach, Ann. Phys. N.Y., 5, 357 (1958);19,287 (1962).
Quasidegenerate Perturbation Theory
The Journal of Physical Chemistry, Vol. 86, No. 12, 1982 2135
Equation 13 serves as a starting point for various perturbation methods. Repeated substitution of (11) yields the Brillouin-Wigner expansion
W = PVP
+ PVGoVP + PVGoVGOp + ...
(14)
Defining “on the energy shell” Green’s functions
which satisfy a Dyson equation similar to (ll),and noting that
G = G’ + G’G(Ej - E )
(16)
equation 12 may be iterated18 to give the RayleighSchrodinger perturbation series
WPj = PVPj
+ PVGojVPj +
PVGoj(VG0’V - C G0’VPLV)Pj + ... (17) KD
103)
Figure 1. Secondorder diagrams contributingto (@.llW1@i).(a) I@/) and I@/)differ by one orbital. (b) I@,) and I@/)differ by two orbitals. The cross in a circle denotes Hartree-Fcck corrections that vanish only if HF orbitals for the state under investigation are used.
This series is the starting point for the general-model-space perturbation theory.18 The derivation of the theory has been described in detailla and will hot be repeated here. The rules for drawing and evaluating the diagrams are given in section 111. B. Model and Shifted Schemes. Two alternative ways to partition the Hamiltonian will be used in the application reported in section IV. The ”model” scheme employs the Merller-Ple~set~~ partitioning Ho = C(ilholi)aifai
(18)
i
v = 1 / 2ijkl2 ( ( i j l U l k l ) U i t U j t U l U k - Ci j( i l f l j ) U ? U j
(19)
where ho is the one-electron Hamiltonian used to generate the orbitals, u is the two-electron repulsion, and f is an average potential added to the one-electron terms of H to form h,,. a and at are the usual annihilation and creation operators. The “shifted” scheme adds the diagonal elements of V to the zero-order Hamiltonian
HOB= Ho + 1/2C((ij[ulij)- ( i j l u ~ i ) ) a ~ a j t a-j aC(ilfli)a?ai i ij
(b)
i
(20) corresponding to the Epsteinn-NesbetB partitioning. Ho8 is the diagonal part of the Hamiltonian matrix in the zero-order basis. The shifted scheme has been used extensively for closed shells. Its application in the general open-shell case has been discussed e1~ewhere.l~ 111. Diagrams A. Single-Block Diagrams. Consider the diagrammatic expansion of the reaction matrix element (ailb l + B j ) , where ai and Gj are model states. The orbital space is partitioned, for the purpose of calculating this matrix element, according to the occupancy in aj: orbitals occupied in aj are holes, other orbitals are particles. Two types of diagrams appear in the expansion, “single block” and “folded”. Single-block diagrams bear close resemblance to the well-known Goldstone diagrams. They are the Goldstone diagrams if the expansion is for the diagonal reaction matrix element ( aiIwai).In the expansion for the offdiagonal element, only diagrams corresponding to the (21) C. Maller and M. S. Plesset, Phys. Reu., 46, 618 (1934). (22) P. S. Epstein, Phys. Reu., 28, 695 (1926). (23) R. K. Nesbet, R o c . R. SOC.London Ser. A , 230, 312 (1955).
(C)
Figure 2. Same as Figure 1. (a) and (b) I@/) and 13)differ by three
-
orbitals. (c) A four orbital difference.
transition 9. ai appear. This means that incoming open lines must be labeled by orbitals occupied in aj but not in ai,and outgoing open lines must be labeled by orbitals occupied in ai but not in aP All incoming lines are classified as hole lines and all outgoing lines are particle lines. The same orbital cannot label an incoming and outgoing line, as it does in Brandow’s formalism. The number of incoming lines is equal to that of outgoing lines; it is simply the number of orbitals describing the transition aj ai. The number of diagrams needed for a particular matrix element is considerably smaller than in previous versions of degenerate MBPT, where all diagrams with up to Nu (the total number of valence electrons) open lines had to be included for every matrix element. All second-order single-block diagrams appearing when aj and aidiffer by 1,2,3, or 4 orbitals are shown in Figures 1 and 2. B. Folded Diagrams. Folded diagrams, first introduced by Brandow,14 appear in all multidimensional diagrammatic expansions. They represent interactions between model states. The general folded diagram consists off + 1 overlapping domains connected by f groups of folded lines (each such group is a “fold”). Each domain is a diagrammatic part, not necessarily linked, representing a transition between two model states. The folded lines entering or leaving the domain are labeled by the proper transition orbitals. Because of the way orbitals are partitioned, the type of the folded line does not always coincide with its direction, and a double-arrow notation is
-
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The Journal of Physical Chemistry, Vol. 86, No. 12, 1982
b
b
d
C
e
l
Figure 3. Folded diagrams derived from the single-block diagram (a). Transitions involving at most two valence orbitals are considered. Diagrams b-d have a single fold, diagrams e and f have two folds. Dotted lines separate different domains.
d
C
x v
x -
f
9
e
h
Figure 5. Illegal folded diagrams. b-h are folded diagrams derived from a. e, g, and h are Illegal because the domains separated by the fold do not overlap. The other folded diagrams are legal.
:s;* - -3
a
a
b
C
Figure 6. Unlinked diagrams appearing in the expansion because of model space incompleteness. (a) The model space includes the determinants (1,2) and (3,4) but not (2,3) and (1,4). (b) and (c) The model space includes the determinants (1,2), (1,4), and (2,3) but not (3,4).
b
d
C
e
f
Figure 4. Open folded diagrams derived from the single-block diagram (a). Diagrams b-f have the same structure as in Figure 3.
adopted. The upper arrow shows the line type (up for particle, down for hole), while the lower arrow gives the line direction, needed in denominator evaluation. Folded diagrams with f folds may be derived by dividing the folded diagrams with f - 1folds into two overlapping parts so that the secondary part (not containing the topmost interaction line) is a legitimate single-block diagram and the other part consists off overlapping domains. Line types are determined by the requirementla that hole folded lines enter the secondary part and particle folded lines must enter the other part, types of previously folded lines are not changed. Line directions remain as in the parent diagram. All the folded diagrams contributing to a given element of the reaction matrix may be derived by this algorithm from the single-block diagrams of the same element, as demonstrated in Figures 3 and 4. The single-fold diagrams in Figure 3b-d are derived this way from Figure 3a, the double-fold diagrams in parts e and f of Figure 3 are ob-
tained from Figure 3d, but the diagrams in parts b and c of Figure 3 cannot be folded further. The encircled domains in Figure 3b-d show transitions from the vacuum or reference state to another model state, while the rest of the diagram represents the transition back to the reference state. The upper encircled domain in Figure 3e and the lower one in Figure 3f correspond to transitions between two model states other than the reference. Open folded diagrams (Figure 4)have the same general structure as their closed counterparts. In all cases, one should be careful to exclude diagrams with nonoverlapping domains, exemplified by parts e, g, and h of Figure 5. The derivation of the folded diagrams for an incomplete model space is somewhat more complicated than in Brandow's'* formalism. In particular, an understanding of the N-electron state associated with intermediate "times" in the diagram may be required. This is a price we pay for the wider applicability and other benefits of the method. C. Unlinked Diagrams, Irreducible Diagrams, and Size Consistency. An unlinked diagram which may be separated into two (or more) parts, each of which is a legitimate diagram appearing in the expansion of a reaction matrix element with the same vacuum (reference) state, is called reducible. All other diagrams (including, of course, all linked diagrams) are called irreducible. The generalized Goldstone expansion may include diagrams which are irreducible but unlinked. At least one part of such a diagram is not by itself a legitimate diagram. Examples are shown in Figure 6. Diagram 6a is in a model space including the determinants (1,2) and (3,4) but not (2,3) and (1,4). The
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The Journal of Physical Chemkitry, Vol. 86, No. 12, 1982 2137
-- -
diagram as a whole represents the model space transition (1,2) (3,4); the separate parts belong to the transitions 1 3 and 2 4, which are not in the model space. Diagrams 6b and 6c are in a model space including the determinants (1,2), (1,4), and (2,3), but not (3,4). One of the parts in each diagram has a model state intermediate, disqualifying that part as a diagram by itself (section IIID); this intermediate is "covered" by the other part, making the whole diagram legitimate. In a complete model space, any combination of the valence orbitals defines a model state, and a separate part is always a legitimate diagram. Unlinked diagrams are therefore reducible and the expansion includes only linked diagrams, as do other degenerate MBPT expansion^'^^" applicable only to complete model spaces. If the model space is exactly degenerate but incomplete, many, but not all, unlinked diagrams cancel. Our previous statementls that all unlinked diagrams disappear for an exactly degenerate model space was in error. It is well-known that unlinked diagrams cause size inconsistency in the diagrammatic expansion for a nondegenerate system. By this statement one means that the value of such a diagram for a system consisting of N identical, noninteracting subsystems is proportional to N", where n is the number of unlinked parts in the diagram. This happens because each such part is by itself a legitimate diagram and may be individually populated by each of the N subsystems. The parts of the irreducible diagrams discussed above are not all legitimate diagrams and cannot therefore be independently labeled. The notion of irreducibility in the generalized Goldstone expansion is therefore equivalent to linkage in the Bruecknel-Goldstone or Brandow methods, and the expansion is always size consistent. D. Summation Rules. Rules for summing diagrams appearing in the expansion for ( 3ilw3,)are collected in this subsection. (a) Ordinary (unfolded) internal hole and particle lines are summed independently over all hole or particle orbitals, respectively. (b) External hole (incoming) and particle (outgoing)lines are summed over combinations of orbitals describing a transition from 3j to 3i. (c) Folds (groups of folded lines) are summed over combination of orbitals corresponding to a model state at the fold. In other words, a domain separated by folds must represent a valid transition between model states. (d) An intermediate state in a single-block diagram may not belong to the model space. (e) A model state intermediate may not appear in a domain of a folded diagram if it is an intermediate state of the whole diagram at the same time (e.g., no model state intermediate may appear between the two topmost interaction lines of diagrams 4c or 5c). If several domains appear between two interaction lines, as between the third and fourth lines from the top of diagram 5b, model state intermediates may not appear between these lines in all domains simultaneously. To determine the intermediate state in a domain, an orbital labeling a folded line entering (leaving) the domain must be taken into account everywhere above (below) its entry point. (0 Exclusion-principle violating (EPV)intermediates are allowed. However, duplicate orbital labels must be counted only once when looking for model state intermediates. (g) The denominator between two interaction lines is given by the orbital energy sum Ccdown
-
-2.65
+b
I
I
I
I
I
I
I
I
I
I
I
- 2.70 -2.75-
0,
E
-2.80 -
c L
0
E - 2.850
W
-2 9 0 -2.951
'/ / I
I
I
I
Flgure 7. Zero-order electronic energy curves for excited 2,' states of He,. n = 2, 3, ....
TABLE I: Basis Seta s: (3360,840,280,93,31.5,10.5,3.5),1.4,0.55,0.22, 0.09,0.03,0.01,0.003 po: (5.4,1.08,0.36), 0.12,0.04,0.012,0.004 p n : (50,10,3.3),1.08,0.36,0.12,0.04,0.012,0.004 d, : 0.048,0.016
Exponents of Gaussian orbitals on each He atom are listed. Exponents enclosed by parentheses belong to a single contracted orbital, with contraction coefficients determined by Hartree-Fock calculations at each internuclear separation. folded lines. The subject of denominator shifts has been discussed e1~ewhere.l~ (h) The overall sign of the diagram is (-lIhfl+f, where h is the number of hole (including folded hole) lines, 1 the number of closed loops after incoming lines are connected to outgoing lines in ascending order of orbital labels, and f is the number of folds.
IV. Quasidegenerate Calculations. Excited Zg+ States of He2 An application of the method presented above to a quasidegenerate system is described in this section. The main object is to show the problems such a system entails and the advantages of the generalized Goldstone expansion in this situation. The states chosen for the demonstration are the c32,+ and CIZg+states of He2 in the region of their potential humps, which result from avoided curve crossing. The zero-order electronic energy curves for excited 2,' states of He2, obtained simply by summing the orbital energies, are shown in Figure 7. The basis set used is given in Table I. The Hartree-Fock equations were solved in the Sil~ e r s t o n e - Y i npotential, ~~ so that la, and la, are Hartree-Fock orbitals of the He2ground state and the others approximate orbitals of singly excited states. The pairs of configurations l a ~ l a , n a , and lagla,2nag exhibit a somewhat unusual behavior.25 They are obviously degenerate at infinite internuclear separation ( E ) ,but they also have a crossing point around 4-5 bohr. This happens because the la, orbital is much lower in energy than la, at small R, but the difference almost vanishes near 5 bohr.
c%p
over downgoing and upgoing lines, respectively, including
(24) H.J. Silverstone and M. L.Yin, J. Chem. Phys., 49, 2020 (1968).
The Journal of Physical Chemistry, Vol. 86,No. 12, 7982
2138
TABLE 11: Shifted Coupling Parameters between I:g+ Configurations of Heza
R
Dl:
D:z
@.I
Of1
Dfz
P i 3
3 ~Symmetry 2 2.9 3.2 3.5 3.8 4.1 4.4 4.7 5.0 10.0
0.36 0.64 1.39 6.86 3.80 1.77 1.26 1.04 1.19
0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.05
2.9 3.2 3.5 3.8 4.1 4.4 4.7 5.0 10.0
0.21 0.38 0.74 1.92 31.82 2.39 1.42 1.09 0.96
0.08
'I:
+
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.06
0.05 0.09 0.15 0.22 0.34 0.49 0.69 0.92 2.46
0.04 0.05 0.06 0.06 0.07 0.08 0.08 0.08 0.01
0.03 0.02 0.01 0.03 0.36 0.30 0.19 0.17 0.19
Symmetry 0 . 1 5 0.07 0.77 0.11 0.52 0.18 0.25 0.26 0.19 0.37 0.16 0.52 0.14 0.71 0.13 0.93 0.27 3.79
0.02 0.01 0.02 0.03 0.04 0.06 0.07 0.07 0.01
0.01 0.01 0.02 0.07 0.27 1.92 0.40 0.28 0.30
0.39 7.53 0.51 0.31 0.24 0.21 0.19 0.18 0.32
a 1 1)= logz10u2u,, 12) = 1 0 ~ l o , ~ 2 013)~=, lag210,30u, and 14) = l o g l o U 2 3 o g .
The nug and nu, orbitals, on the other hand, remain well separated up to much larger R, because of their greater spatial extent. The avoided crossing of the n = 2 curves, which occurs at 4 bohr, leads to considerable humps in the c3Z,+ and CIZg+states of He2. A crucial decision to be made in the application of any quasidegenerate perturbation theory involves the construction of the model space. A simple criterion may be derived by considering a two-state space, with the diagonal matrix elements E; = (ilHli),i = 1 , 2 , and the interaction Vlz = (llH12). Defining the coupling parameter
the solutions of the two-state secular equation are
E = '/2[ElS+ Ezs + (Ela- E2')(1
+ 4&,2)1/2]
(22)
These solutions may be expanded in convergent power series starting with E," and Ezsif P12 C 0.5. A similar series is obtained in the diagrammatic expansion if only two states are considered.% The situation is, of course, more complicated in a real case with many states, but coupling parameters defined as in (21) can still serve as guidance in choosing the model space. Note that (21) applies to the shifted scheme; an equivalent definition in the model scheme is obtained upon replacing E: by Ei. It is interesting to note that the series expansion of (22) includes only even powers of 0. This is related to the fact that no third-order diagrams appear for the simple twostate model if the shifted scheme is employed (diagonal third-order diagrams exist in the model scheme). Rearrangement diagrams' appear, however, in the fourth order. Third-order convergence may not prove therefore that the series is indeed converged. If fourth-order corrections are not calculated because of the costs involved, an estimate of the convergence may be obtained by comparing model and shifted third-order results, as the latter include diagonal fourth- (and higher-) order corrections. (25) S.L.Guberman and W. A. Goddard, Chem. Phys. Lett., 14,460 (1972); Phys. Rev. A , 12, 1203 (1975). (26) Note that the total space (not the model space) includes only two states.
Hose and Kaldor
Coupling parameters of four Hez Zg+ configurations at different internuclear separations are collected in Table 11. They were calculated in the shifted scheme, but model scheme parameters are similar in magnitude. The two lowest confiiations are strongly coupled above 3.5 bohr; coupling with the other configuration is weaker. The first set of calculations was performed for the two triplet determinants, (lu,21uua2uua)and (lugala,22u ), each taken separately as a single-state model space. able I11 shows the model scheme corrections to fourth order, except for the fourth-order triple excitations calculated at two points only. An interesting feature of these results is that they appear to be converged at third order (hE3is much smaller than hEzexcept for R = 4.1 bohr, very close to the crossing point). Fourth-order corrections are, however, large in many cases, showing the convergence is illusory. The same behavior is exhibited by the shifted-scheme energies in Table IV. Third-order corrections are very but many fourth-order values are large (note that only the double-excitation term AE4d is calculated, and there is partial cancellation with the other terms). Both schemes show poorer convergence the larger Plz is. Even without calculating fourth-order corrections, one could infer the poor convergence from the erratic behavior of the energy corrected to third order as a function of R and from the substantial differences between model- and shifted-scheme results (Table V). Single-state model spaces (or nondegenerate MBF'T) are obviously inadequate for this system. Quasidegenerate calculations, implementing the theory presented in sections I1 and 111, were therefore performed for the c3Zg+and CIZg+states. The following model spaces were used: (a) the single configuration lu,21u,2au for R I 3.2 bohr; (b) a two-dimensional space, adding the configuration luglu,22u ,for 3.5 I R I 4.7 bohr; (c) a three-dimensional space, adding lug2u,23a ,at R = 5 and 10 bohr. The largest model space comprised therefore three configurations or six determinants. Transitions between model states involved one or two orbitals, requiring diagrams with zero, one, or two open lines (a given matrix element involves only diagrams with a constant number of open lines, corresponding to the transition represented by the element). A complete-model-space m e t h ~ d ' ~ Japplied ' to the same problem would involve 25 lZg+ configurations or 52 determinants, and each matrix element would require diagrams with up to four open lines. The results obtained with quasidegenerate model spaces are summarized in Table VI for the singlet and Table VI1 for the triplet. In addition to the model spaces defined above, they include calcualtions at 2.9 and 4.7 bohr with a larger space to test the effect of adding an extra state. The results show good convergence, small differences between two model states at the two points where such comparison was made, reasonably smooth dependence on R , and good agreement ( I 6 mhartree) between the model and shifted schemes. The coupling parameters (Table 11) indicate that a somewhat larger model space than used in this preliminary application is probably required at certain internuclear separations. A more complete investigation is in progress and will be reported elsewhere.
fl
V. Summary and Conclusions The diagrammatic perturbation theory presented in this paper is applicable regardless of the structure of the (27) The better convergence of the shifted scheme for the excited states may be traced to the infinite summation of Hartree-Fock corrections, usually absent from ground-state calculations. These corrections appear because the orbitals are not HF orbitals of the state under consideration. See ref 19 for a more detailed discussion.
Quasidegenerate Perturbation Theory
The Journal of Physical Chemistry, Vol. 86, No. 12, 1982 2139
TABLE 111: Energy Corrections (hartree) Obtained in the Model Scheme for the Two
R
AE,
AEI
AE,
AE3
Determinants Shown" A E , ~
A~ , d
2.9 3.5 3.8 4.1 4.4 4.7
-3.428 -3.198 -3.108 -3.031 -2.964 -2.906
077 448 682 627 936 733
-0.123 672 -0.156 381 -0.246 681 +0.154 148 -0.042 966 -0.078 507
( log2,l o ,a,20 ua ) +0.013 926 -0.011 764 t0.010346 -0.036 846 -0.315 795 -17.240 073 -0.027 917 -0.009 566
-0.020 -0.107 -0.825 -2.808 -0.282 -0.152
215 826 478 003 127 766
2.9 3.5 3.8 4.1 4.4 4.7
-3.430 -3.201 -3.112 -3.036 -2.970 -2.912
400 869 770 380 272 494
-0.106 191 -0.083 775 +0.011 515 -0.390 003 -0.196 743 -0.166 982
( 10ga,luU~,2uga) +0.038 643 t0.021 623 +0.063 527 t0.339 650 +0.049 266 +0.028 502
-0.163 -0.093 -0.699 -4.172 -0.427 -0.205
202 990 722 552 310 197
-0.005 218 -2.060 686
A E , ~ t0.013 669 +0.185 838 +3.001 547 -12.371 384 t0.047 210 t 0.1 50 082 +0.258 521 +0.104 534 -0.650 143 +23.102 256 + 1.008 551 t 0.365 163
Single-state model spaces. The superscripts d, t, and q denote double, triple, and quadruple excitations, respectively. TABLE IV: Shifted Energy Corrections (hartree) for the Two
2.9 3.5 3.8 4.1 4.4 4.7 a
-0.108 154 -0.146 737 -0.337 105 +0.023 978 -0.051 375 -0.076 654
-0.006 -0.007 - 0.009 -0.007 -0.008 -0.009
556 461 229 199 306 034
3 x 2 Determinants Showna
-0.049 841 -0.060 713 +0.132 244 -0.230 691 -0.160 181 -0.142 527
-0.011 069 - 0.105 346 -2.167 427 -0.604 538 -0.139 706 -0.093 635
161 732 230 367 373 752
- 0.700 432
-0.082 -2.088 -0.750 -0.185 - 0.104
542 318 988 852 000
Single-state model species.
TABLE V: Third-Order Electronic Energies lhartreel for the Two
3x
+
Determinants Showna
(logs,1U U 2 , 2oga )
(1og2,luua,2uua)
R
E3
E3
2.9 3.5 3.8 4.1 4.4 4.7 a
-0.008 -0.004 -0.004 -0.007 -0.007 -0.007
-6.445 -6.220 -6.259 -6.053 -5.891 -5.846
962 996 395 680 229 482
-6.450 -6.229 -6.322 -5.875 -5.880 -5.855
-6.313 -6.112 -5.895 -5.951 -5.987 -5.923
926 158 202 254 027 097
498 759 845 408 099 720
-6.303 951 -6.116 051 - 5.842 873 -6.139 113 -6.007 177 -5.935 519
Single-state model spaces. E , and E,S are in the model and shifted schemes, respectively.
TABLE VI: Total Energies (in hartree) for the C'x,C State of He, 2.ga 2.gb 3.2a 3.56 3.8b 4.1b 4.4b 4.7b 4.7c 5.0'
lo.oc
-4.947 -4.951 -4.934 -4.935 -4.932 -4.932 -4.932 -4.932 -4.937 -4.937 -4.940
Model space logzlo,20,. logla,f30g
265 337 177 409 847 278 419 705 118 849 324
-5.066 852 -5.066 216 -5.058 013 -5.050 222 - 5.048 213 -5.048 712 - 5.050 778 - 5.054 082 - 5.047 081 - 5.047 736 -5.045 894
-5.049 276 -5.048 139 - 5.040 191 - 5.034 053 -5.031 048 - 5.030 185 -5.030 195 -5.030 536 -5.034 293 - 5.035 441 -5.041 601
-5.048 683 -5.047 590 - 5.040 274 -5.031 915 - 5.030 058 -5.030 842 - 5.033 246 - 5.036 874 -5.030 379 - 5.032 170 -5.031 407
-5.053 617 -5.053 332 -5.043 981 - 5.039 001 -5.036 105 -5.035 326 -5.035 419 -5.035 859 - 5.038 526 - 5.040 283 -5.043 571
Model spaces l o g z l o , 2 ~ , and l a g l ~ u z 2 0 g . The two determinants in footnote b and
zero-order function space, which may be nondegenerate, quasidegenerate, or degenerate. As in all perturbation theories, the Hamiltonian is diagonalized within a lowdimensional space (the model space), and interactions outside it are included perturbatively. The novel feature of the present method is the selection of model states on an individual basis, after examining their interaction with other states. This is to be contrasted with other versions of degenerate MBPT, all of which assume complete model spaces, including all possible distributions of electrons in the open shells. Such spaces may be large and wasteful, as many of the states thus constructed are of little or no
importance. The energy range spanned by such a space may be quite broad, increasing the danger of intruder states which spoil the convergence. Another departure from previous methods is the partitioning of orbitals into two classes, holes and particles, instead of the usual three (core, valence, and particles). This partitioning leads to reference states closer to the physical situation, to a significant reduction in the number of diagrams to be evaluated, and to a greater degree of diagram cancellation before the calculations. Certain unlinked diagrams appear in the expansion if the model space is incomplete. They are all irreducible,
J. Phys. Chem. 1982, 86, 2140-2153
2140
TABLE VII: Total Energies (in hartree) for the C 3 x i State of He," R 2.9" 2.gb 3.2" 3.5b 3.86 4.1b 4.4b 4.7b 4.7c 5.OC
lo.oc a
E, -4.956 -4.967 -4.943 -4.955 -4.955 -4.955 -4.956 -4.957 -4.958 -4.959 -4.956
906 171 340 835 018 537 422 229 290 044 493
E, -5.080 578 -5.078 896 -5.077 050 -5.073 864 - 5.071 475 -5.071 899 - 5.073 656 -5.076 490 -5.073 301 - 5.074 640 -5.077 370
E3
- 5.066 652
-5.063 397 - 5.063 762 -5.058 174 -5.058 888 - 5.060 053 -5.061 511 - 5.062 853 - 5.063 995 - 5.065 376 -5.073 769
E*S - 5.064
574 -5.064 201 -5.061 286 -5.060 165 - 5.058 203 - 5.059 083 -5.061 274 - 5.064 475 -5.061 262 -5.063 7 7 1 -5.066 537
-5.071 727 - 5.067 353 -5.069 618 -5.063 560 -5.063 918 -5.064 882 -5.066 176 - 5.067 416 -5.068 890 - 5.071 499 -5.078 314
Model spaces are described in footnotes a-c in Table VI.
i.e., they cannot be separated into parts which are by themselves legitimate diagrams. The expansion is therefore size consistent. The method was applied to the c38,+ and C?8 states of Hez near an avoided crossing of the potentiaf curves. While nondegenerate MBPT diverges in this case and +
other degenerate theories require a 25-configuration model space with up to four-open-linediagrams, good convergence was obtained by using the present method in a threeconfiguration space with diagrams having no more than two open lines. Applications to other atomic and molecular systems are under way.
Newtondaphson Approaches and Generalizations in Multlconfigurational Self-Consistent Field Calculations Danny L. Yeager,' Dlane Lynch, Jenrey Nlchols, Department of Chemistry, Texas A&M Universw, College SiWion, Texas 77843
Poul Jerrgensen, and Jeppe Olsen Chemistry Department, Aarhus UniverMy, OK 8000, Aarhus C, Denmark (Received: July 13, 198 1)
We develop and review multiconfiguration Hartree-Fock procedures based on unitary exponential operators. When the energy functional is-expanded through second order in the orbital optimization operator, 2, and the state optimization operator, S, and the variational principle applied, the resulting set of coupled, linear inhomogeneous equations are known as the Newton-Raphson equations. These equations demonstrate quadratic convergence. Particularly when far from convergence, small eigenvalues of the Hessian may give extremely large step length amplitudes. We discuss two procedures, mode damping and mode controlling, which are used to reduce large step length amplitudes. With both procedures mode reversal is also used to assure that the proper number of negative eigenvalues is present for each iteration. New calculational results are given for the first excited lZg+ state of Cz and the 21A1 state of CH2. For an optimized MCSCF state the generalized Brillouin's theorem is satisfied. There may be other criteria which we want the MCSCF state to fullfil, e.g., the nth excited state of a certain symmetry have n negative eigenvalues of the Hessian. We discuss several such criteria, including that the MCTDHF not be unstable and that the MCTDHF should have n negative excitation energies for the nth excited state of a certain symmetry. Finally, generalizations of Newton-Raphson and the multiplicity independent Newton-Raphson (MINR) approaches are discussed. We show how fixed Hessian-type approaches will demonstrate quadratic, cubic, quartic, ... convergency. Calculations are presented for the E3Z; state of Oz. MINR approaches may be useful when small eigenvalues of the Hessian are present. Since the MINR formula involves the product of third-derivative matrices with vectors, explicit evaluation of the third-derivative matrices is avoided.
I. Introduction Recently, multiconfigurational self-consistent field (MCSCF) techniques have received considerable, renewed interest.'-14 This is in large part due to developments (1)A. C. Wahl and G. Das, "Modem Theoretical Chemistry", Vol. 3, H. F. Schaefer, 111, Ed., Plenum Press, New York, 1979, Chapter 3. (2)E. Dalgaard and P. Jsrgensen, J. Chem. Phys., 69,3833 (1978). (3)D. Yeager and P. Jsrgensen, J. Chem. Phys., 71,755 (1979). (4)E. Dalgaard, Chem. Phys. Lett., 65,559 (1979). (5)D. Yeager and P. Jsrgensen, Mol. Phys., 39,587 (1980). 0022-365418212086-2140$01.25/0
involving Newton-Raphson kchniques2-8and generalizations.13 These ~ r o c e d u r e s ~usually -'~ show superior con(6) D. Yeager, P. Albertsen, and P. Jsrgensen, J.Chem. Phys., 73,2811 (1980). (7)P.Jsrgensen, P. Albertsen, and D. Yeager, J. Chem. Phys., 72,6466 (1980). (8)C. C.J. Roothaan, J. Detrich, and D. G. Hopper, Int. J.Qwntum Chem. Symp., 13,93 (1979). (9)P.Siegbahn, A.Heiberg, B. Roos, and B. Levy, Phys. Scr., 21,323 (1980). (10)B. Lengsfield, 111, J. Chem. Phys., 73,2342 (1980).
0 1982 American Chemical Society
