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The Reduced Partitioning Procedure in Configuration Interaction Studies By RODNEY JOSEPH BARTLETT A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE tIr'I RSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE PCQ.'IRL.LMN:TS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1971 DEDICATED to MY PARENTS ACKNOWLEDGE E; I NTS It is my pleasure to gratefully acknowledge my chairman, Prof. Yngve Ohrn, for the unfaltering assistance, encouragement, and supervision he has given me during the period I have spent in the Quantum Theory Project. Without his cooperation in all manner of things, my stay here would have been much less rewarding. Also, I wish to sincerely express my apprecia tion to Prof. PerOlov Lwdin, my cochairman, for his intellectual stim ulation and for several conversations in which he made numerous fruitful comments which improved the quality of this dissertation. To Dr. Erkki Br2ndas, I owe a special word of thanks, since this research was undertak en with his assistance and would have been impossible without his guidance. To say that the exchange of ideas between us was of great benefit to me is an understatement. I also wish to express my appreciation to the other members of my supervisory committee for their continued cooperation. All of the members of the Quantum Theory Project have been helpful to me, but particularly I would like to thank Dr. C. E. Reid and Dr. Joe Kouba for assistance in the use of some of their computer programs, as well as Nelson Beebe and John Bellum for additional computational aid. I am also indebted to Dr. Micheal Hayns for his careful drawing of the figures. Lastly, I am especially grateful to Beverly Bartlett for her tireless efforts in the typing of this dissertation as well as her constant encour agement during its preparation. TABLE OF CONTENTS AC ,O'.ILFDGEMENTS . . . LIST OF TABLES . . . LIST OF FIGURES . ... . . ABSTRACT . . .. Chapter I. INTI ODiJCTION . . . II. THE PARTITIT.O;IiG TECHNIQUE AND PERTURBATION THEORY II.1 The Partitioning Technique . . 11.2 The Partitioning Technique in Operator Form 11.3 BrillouinWigner Perturbation Theory . 11.4 RayleighSchr'"dinger Perturbation Theory . 11.5 Inner and Outer Projections . III. PADE A :.'. LA TS . . 11.1 Derivation and Definitions . 111.2 Connection with Moment Expansions and Inner Projections . . IV. THE REDUCED PARTITIONING PROCEDURE . IV.1 NonLinear Summation of Perturbation Expansions . . IV.2 A Model Hamiltonian for a Finite Basis . IV.3 The Fundamental Equations for the Reduction Process . . . IV.4 'i. Method of Steepest Descent and the Coometric Sumrule . . Page . iii . Vi * vii . .. .viii I I I I V. EXCITTED STATES . . 50 V.1 The SuperMatrix Formulation . 50 V.2 The Selective Reduction Procedure . 56 V.3 The Geometric Formula for a MultiDimensional Reference Space . . 59 VI. NUMERICAL APPLICATIONS OF THE REDUCED PARTITIONING PROCEDURE . . 62 VI.1 Summary of the Computational Procedure 63 VI.2 Results and Discussion: The Single Partitioning Case . 69 VI.3 Results and Discussion: The Selective Reduction Procedure for Multiple States 87 BIBLIOGRAPHY . . . ... 106 BIOGRAPHICAL SKETCH . . . 110 LIST OF TABLES Table Page 1. Reduction Energies for H2 with the Coulson Reference Function . . 71 2. Reduction Energies for H2 with a HartreeFock Reference Function . .. 73 3. Reduction Energies for HeH with a HartreeFock Reference Function . . 75 4. Reduction Energies for the X E State of HeH as a Function of Internuclear Separation . 77 5. Spectroscopic Parameters for Different Approximations to the X17 State of HeH . 85 6. Illustration of Selective Reduction for Multiple States of Hel . . 88 7. Reduction Energies for the A Z+ State of HeH as a Function of Internuclear Separation . 91 8. Comparison of Rigorous and Approximate Multiple State Reduction Energies for HevH as a Function of 97 Internuclear Separation . . 9. Reduction Energies for the 1 Excited States of HeHl as a Function of Internuclear Separation . 101 LIST OF FIGURES Figure 1. A Comparison of the HF and Full CI Potential Curves for the Ground (X1L+) State of Hell . 2. Low Order Reduction Energies for the X E State of He' . . ... 3. Full CI Results for the Ground and First Excited States of HeH . . 4. Low Order Reduction Energies for the A E State of HeH . . . 5. Improved Excited E1 States for HeH . 6. Reduction Energies for the BI E State of HeH . 95 100 104 * O Page * . . . Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE REDUCED PARTITIONING PROCEDURE IN CONFIGURATION INTERACTION STUDIES By Rodney Joseph Bartlett June, 1971 Chairman: Prof. Yngve Ohrn CoChairman: Prof. PerOlov LVwdin Major Department: Chemistry Within the configuration interaction (CI) framework, a higher order perturbation approach has been developed which allows a drastic reduction in the number of nparticle functions required to give the energy of a CI wavefunction to a desired numerical accuracy. By means of this method, significant improvements over the standard HartreeFock (HF) and truncated CI calculations are obtainable. Since the procedure does not require solutions of large secular equations, the technique also has implications for full CI calculations. As this procedure complements the natural orbi tal methods, the two techniques can be used in conjunction for additional flexibility. The theory derived to effect the reduction process, i.e., the reduced partitioning procedure, employs the techniques of inner and outer projec tions and Pade approximants to yield a tractable computational scheme with known convergence properties. The theory is formulated both for a single unperturbed reference function and for a multidimensional reference space to use in applications to excited states. In the latter case, a method viii which uniformly treats the excited states as well as one which permits a selective reduction for one specific state is obtained. These reduction equations are also shown to be related to the method of steepest descent. The first order solution is of particular significance. In the examples studied, this result, which corresponds to the application of a geometric sumrule to the perturbation expansion, is found to give sixty to seventy per cent of the possible energy improvement obtainable from the specified basis. This solution also has some computationally simpli fying features when the reference function is of the HF or CI variety. Preliminary applications to the H2 molecule and the HeH molecular ion tend to justify the reduction procedure for more general cases. In these examples, no more than a twelfth order solution is required to give essential agreement with the full CI result in each case, even though as many as eightytwo configurations constitute the full nparticle basis. In addition, it is found that only a third order solution is required to account for more than ninety per cent of the possible correlation improve ment over a HF calculation. These calculations presented here also in clude potential curves for the HF, the full CI, and the perturbation corrections for selected states of HeH . CHAPTER I. INTRODUCTION The majority of theoretical studies of atoms, molecules, and solids has its origin in the time dependent, nonrelativistic Schr6'dinger equa tion [I], HI = ifi (1.1) 6t More commonly, though, one focuses on its progeny, the timeindependent form HY = E Y (1.2) n n n which defines the "stationary state" eigenfunctions, ([Y The complete Hamiltonian operator for a given problem is an enormously complex entity, whose eigenfunctions would contain all electronic, vibrational, and rota tional motions that are possible for the system as well as a variety of hyperfine interactions. This enigma is never considered in its entirety, but, subject to the problem of interest, one makes certain assumptions as to the form of H. Even then the equations obtained are probably still partially coupled and require another approximation to effect a separation. It is this process and the BornOppenheimer, or "clamped nuclei ," approxi mation [2] that lead us to the usual Hamiltonian for the electronic motion, H = E2 + 1/r.. > Z /r + E Z Z /R (1.3) 1. i The quantities in (1.3) are the kinetic uaL.';/ operator and terms arising from the couloibic potentials for electronelectron, electronnuclear, and 1 nuclearnuclear interactions. Within the clamped nucleii approximation the last term is merely a constant. When additional interactions are to be considered, they are customarily treated as a perturbation of (1.3). Even though (1.3) is a distinct simplification, it is still impossible to solve (1.2) for a system with more than one electron, necessitating approximate solutions as the only recourse. In this respect, powerful methods have been developed to approximately solve (1.2). These fall pri marily into two categories, variational methods, based on the Ritz varia tional principle [3], and perturbation theory. The former has tradition ally been the more important of the two, with the latter essentially limited to the effects of external fields. However, a renewed interest in perturbation methods is very much in evidence currently. Since (1.3) is bounded from below, the variational theorem allows one to approximate Y by some Y which contains several parameters to be opti mized, form the functional = <(_lHji)/<(i> ,) (1.4) and by minimizing E with respect to the parameters embedded in % determine an approximate wavefunction and an E E. With a sufficiently flexible expression for Y, one can thusly obtain as good a solution as desired. In this manner the energies of a few simple twoand threeelectron systems have been calculated to experimental or better accuracy. For less simple systems, two relatively general variational methods have replaced the "special case" solutions of the previous period. These are the HartreeFock method [4], which for molecules is invariably imple mented as formulated by Roothaan [5], and the method of superposition of configurations or configuration interaction. The ordinary HartreeFock (HF) wavefunction for an atom or molecule with n electrons is assumed to be a single Slater determinant, HF =((1)(2(2) ... *n(n)) (1.5) composed of n oneparticle functions (spinorbitals) where each spin orbital is a spatial orbital i. associated with either a! or P spin. In the Roothaan procedure, the functions ({.3 are assumed to be approximated by expanding in a given set of functions [X }, Z = E X c (1.6) with arbitrary expansion coefficients c The coefficients can be determined by the variational principle (1.4) to give the lowest possible energy for the single determinant form. Besides being conceptually satis fying, a wavefunction of this form possesses a degree of justification for the prediction of ionization potentials and oneelectron properties due to the Koopmans [7] and MoellerPlesset [8] theorems. On the other hand, all single determinant wavefunctions are unable to properly treat the instantaneous electron interactions, i.e., "correlation," which arises from the second term of (1.3). This has several consequences, one of which is that a standard HartreeFock wavefunction for a molecule cannot general ly be used to obtain reasonable potential energy curves since it will not dissociate properly except in a few very special cases. Furthermore, one encounters difficulties with symmetry in proceeding to open shell studies resulting in the near impossibility of treating excited states. The configuration interaction (CI) procedure effectively eliminates the complications inherent in the HartreeFock scheme, but not without *A common basis set is the Slater type orbitals [6], X(l,m,n) = n1 a'r r e Y(l,m), where l,m, and n are the usual quantum numbers., r the mag nitude of the radial coordinate relative to an arbitrary origin, Y(l,m) the spherical harmonics, and a the orbital exponent which is to be selected in some manner. causing some new problems of its own. If one considers the expression (1.6) and if there are m expansion functions [X 3, then with spin, it follows that 2m(i.) are possible. Consequently, if 2m > n, many more than one determinant may be generated from these ({.* by forming the r3 ( ) combinations. In the CI method one can consider these determinants, n [Dk}, to be the basis for a trial wavefunction, E D kCk (1.7) k with the (Ck) to be determined from (1.4). Other than the multideterminantal nature of the CI approach, which permits electron correlation, the primary distinction between it and HartreeFock theory is that the expansion coeffi cients in the ([.j are not usually determined via an energy criterion but rather fixed  such as by orthogonalization of the Nx )  thereby placing the flexibility of the wavefunction exclusively within the [Ck}. (Exclud ing the multiconfigurational HartreeFock techniques, the one exception to this statement involves natural orbital iterations as discussed below.) Immediately, a problem with the expression (1.7) becomes apparent, 2m since the number of determinants ( ) rapidly becomes astronomical even n for a relatively modest basis set, (x }. If one is interested in a state or states of a certain symmetry, many of these determinants can be excluded by group theory since they will contain no component of that state. Even better, the individual determinants which do contribute to the state of interest can be grouped into linear combinations, (BL such that each BL is a pure symmetry function. By so doing, the expansion (1.7) can be substantially shortened, but,even so, the number of configurations is still likely to be very large. If one forms the approximate wavefunction S= BLCL (1.8) L by including every configuration that can be constructed from the given basis (X 1, then the result obtained from the variational principle is termed the "full" CI solution for the problem. This solution has some convenient properties. First, it is invariant to any transformation among the basis functions {X } and thus is independent of the explicit form for the [(i.; or, equivalently, the final result is solely determined by the initial basis (X 3. The second advantage is that by using all pos sible (BL}, no dubious selection of configurations needs to be made. Un fortunately, though, the number of configurations needed for the full CI is still usually much too large to enable one to use all of them, a fact which leads to severe diagonalization problems among others. In order to avoid most of these problems, it is necessary to make a somewhat arbitrary selection of the configurations thought to be important and then solve this "truncated" CI problem. If so, the full CI features are no longer generally true. In particular, the explicit combinations [.} are impor tant to the results obtained. A second problem encountered, the choice of basis (X }, is a universal one for all methods with the exception of purely numerical solutions. In some ways, however, the selection is more crucial for CI studies than in others, such as the HF. The reason for this is twofold. First, contrary to a HF calculation, if the CI procedure is employed, one is generally interested in excited states as well as the ground state. This usually requires a basis that can do both adequately, a task which is e:.:cec'dingly difficult to manage. Also, there is little or no "chemical intuition" which can u;u:' ;t orbital exponents that will allow a satisfactory de scription of excited states. Of course, if the basis set were large enough, then the choice of the (X } would not be as crucial, and this is a fairly reasonable alternative in HF theory, but in CI one is again led to the problem with the number of possible configurations. One has essentially attempted to resolve this dilemma with two meth ods, both used separately and in conjunction. These are orbital exponent optimization, which is most effective when undertaken for each state in dependently, and natural orbital iterations. The first of these is self explanatory, although still largely prohibitive due to the extreme cost. The second is a potent tool for improving less than full CI calculations by exploiting the flexibility in the selections of the linear combinations (i}. This technique makes use of the concept of "natural spin orbitals," introduced by Lowdin [9], which are defined as the eigenvectors of the first order reduced density matrix, y(xIx') = ? (x x2...xn) Y(x'x2 ..x )dx2...dx (1.9) in the sense of jY(xJx)cpk(xp)dx = nkk(xl) (1.10) where nk is the occupation number associated with the natural spin orbital ckp. From a theorem due to Schmidt and later Coleman [10], it is found that this set of orbitals has a certain optimum property. That is, for a full CI wavefunction, the selection of the natural orbitals with largest occu pation numbers as the spin orbitals from which the configurations are constructed results in a much shorter approximate expansion which has the property of being in maximum possible coincidence with the full CI [11]. Natural orbital iterations further utilize this feature, except now per taining to a truncated CI. Since such an approximation is not invariant to the choice of the ({i), by obtaining a set of approximate natural orbitals from the density matrix associated with this CI wavefunction, one can select a set via an occupation number criterion from which another trial CI wavefunction is generated. This set is expected to be an improve ment over the previous ({.}. Consequently, by continuing the process and assuming convergence, the mixing of the basis functions (X 3 is thusly "optimized" [12,13]. This technique is very useful, but it also has de ficiencies, one of which is the dependence on the initial selection of configurations. Having outlined the essential components of the two most important quantum chemical methods, given a satisfactory basis set, the predominant problem that emerges in progressing beyond the HartreeFock approximation, as well as utilizing configuration interaction to maximum efficiency, lies in the exorbitant numbers of configurations that need to be considered. The treatment of this standard problem in various contexts is the objective of this investigation. By appealing to the partitioning technique [14,15] and perturbation theory, we shall develop an approach to this problem that differs from the natural orbital methods, although it is entirely com plementary. This process, the "reduced partitioning procedure," shows promise of being useful in a variety of applications within the configu ration interaction domain. The basic idea of the procedure is to reduce the dimension of a CI problem by grouping N configurations into M functions, such that the energy can be obtained to a desired numerical accuracy even though M << N. The main distinction between this "reduction" procedure and the natural orbi tal methods is that the M functions obtained are linear combinations of the previous configurations, whereas a natural orbital "reduction" replaces one set of confi.,~I' ILitio; with a second. Thus, since each to.chiiiique em phasizes different aspects of the problem, they can be used in conjunction for additional flexibility. 8 The method employed to achieve this result is essentially perturbation theory, but in a more general framework than is customary. Traditionally, the feasibility of applying perturbation theory to problems has depended on three elements: the existence of an exact solution to an unperturbed Hamiltonian; the convergence of a linear perturbation expansion; and, due to inherent computational problems, the suitability of first and sec ond order solutions. These are impractical restrictions for arbitrary, manyelectron systems, however, and preferably variational principles should be incorporated with perturbation techniques to give more powerful methods for the calculation of molecular properties and energies. The perturbation approach we have formulated includes the possibility of using any approximate wavefunction as the unperturbed solution, guarantees con vergence via nonlinear summations, and allows higher order solutions to be easily obtained. The reduced partitioning procedure (RPP) will have several advan tages over the more standard methods. For instance, a full CI calcula tion is seldom soluble since the diagonalization of large Hamiltonian matrices, extremely time consuming at best, is usually impossible due to near linear dependencies. Although the reduction process would generally require that all the matrix elements are constructed, it would only require a diagonalization of an M x M Hamiltonian matrix for M << N, thereby pro viding a feasible alternative to the full CI solution. Even though a full CI may not be the objective, one may be interested in certain intermediate solutions which would give a substantial improvement over a HartreeFock or truncated CI wavefunction. In this respect, also, the RPP suggests some possibilities. One of these, the first order solution, yields a dramatic energy decrease while possessing important computational simpli fications. This first order solution can also be connected with the "geometric" approximation and the method of "steepest descent." For excited states, the RPP will provide two alternative approaches which have different advantages. One of these allows a selective treatment of the particular state of interest. The question of properties other than the energy is also a significant one, since an ordinary perturbation treatment of second order properties from a CI perspective would require a set of N1 excited states, or, equivalently, the inversion of an N1 x N1 matrix. Consequently, here also, a systematic reduction of the space is warranted, as this would permit a reasonable approach within the CI framework. After presenting background material on the partitioning technique, perturbation theory, and Pade approximants in the next two chapters, the general formulation of perturbation theory and the reduced partitioning procedure will be developed in Chapter IV, with the discussion of excited states following in Chapter V. In the final chapter, numerical applica tions to the 112 molecule and the HeH molecular ion will be described. CHAPTER II. THE PARTITIONING lECHNiQUE AND PERTUR'BATION THEORY The partitioning technique, as developed by LUwdin [14,15], has been shown to be of primary importance in elucidating the interrelationship between perturbation theory and the eigenvalue equation as well as leading to a theory of upper and lower bounds to energy eigenvalues. In the fol lowing, the partitioning technique will be briefly presented from which the resolvent formulation of perturbation theory will arise as a logical consequence. In addition, the important idea of inner and outer projec tion will be described in the last section. II.1. The Partitioning Technique The paramount objective of molecular quantum mechanics is the solution of the stationary state Schr'dinger equation, HY = EY (2.1) where H is the Hamiltonian for the system and Y an eigenfunction. If we assume an expansion for Y in terms of an orthonormal nparticle basis fE) = Iflf...f 2 n..), then instead of (2.1) we obtain the matrix form of the eigenvalue problem 1BC = G E (2.2) where the expansion coefficients for each state are grouped into I = ( n ...). If i is complete, then (2.1) and (2.2) are equivalent. For a finite basis, assumed to be of the order N+I1, the eigenvalues 10 (EE ...E ) are respectively upper bounds to the exact solutions. By defining M = H 8*1, (2.2) can be written M = 0 (2.3) for M = <(f MIff). If we now consider IfE)to be partitioned into two subsets ja )and[lb) of order na and nb, respectively, then it follows that a b Ml = MI aa ba ab Mbb (2.4a) From (2.3) and a0 al" an bO T bl bn (2.4), we have the system of equations NI C + C , aa ai ab bi ba ai bb bi (2.5a) (2.5b) for each i. By solving (2.5b) for bi one obtains Cbi = 1 b ai bi bb ba ai (2.6) 1 as long as Mb exists. The substitution of (2.6) into (2.5a) gives the expression (M M M I ) M = C ,D aa ab bb ba ai a or, equivalently, if written in terms of the Hamiltonian matrix, [(11 ) + (& 1bb (11 ) )bb) ] ba ]C a = (D aa aa ab bb bb ba ai (2.7) . (2.8) a (2.4b) Consequently, by defining the modified Hamiltonian matrix )1 aa aa ab bb bb ba (2.9) we have the altered eigenvalue problem S= S ., (2.10) aa ai ai 10 which yields n roots for F = P. = E.. a 1 1 The choice of partitioning is completely arbitrary, and, in the course of this work, various choices will be made. For the present, however, let us restrict ourselves to the case when n = 1, and call this single func tion f = c with C 0 0. From (2.10), we have = H = H + H (1 H ) (2.11a) o oo oo ob bb bb) bo' (2.11a) or, with E = H =. .(polcPo )l 8 = Eo + (oHj Ib) (1 bb ( bj H Ib)) 1(b H ) (2.11b) This implicit function of F, (2.11), is the "bracketing function" which is crucial to the theory of upper and lower bounds [15]. The terminology comes from the fact that for any e > E then 8 < E which allows one to So o "bracket" the eigenvalue E . o The connection between (2.11) and BrillouinWigner [16] perturbation theory could be obtained directly by separating the inverse matrix into its diagonal and nondiagonal parts and then expanding with respect to the latter by means of the relation (A A)1= A1 1 1 1 1 1+ ....(2.12) (A (B)" = A" A ^ A + A IBAIBA ....(2.12) The relationship with perturbation theory, however, is more conve niently presented in terms of the operator formulation of the partitioning technique. 11.2. The Partitioning Technique in Operator Form In order to discuss the operator formulation it is necessary to in troduce the projector S= oo) (' (2.13) which projects the reference function po from the space of interest. oD: idempotent and selfadjoint, i.e., &2 = f, = e0, (2.14) where W' indicates the Hermitian adjoint. In addition, the projector for the orthogonall complement" to cp the space previously spanned by  b), is P = 1 0", (2.15) with P2 = P and P = P as before. Also, since aP = PO'= 0, (2.16) the projectors U and P are said to be "mutually exclusive." In order to treat (2.1) we shall also need the definition H = PHP (2.17) which is an "outer projection" of H with respect to the subspace of P (see 11.5). The outer projected Hamiltonian (2.17) satisfies the relation Hik= EkPk (2.18) where Yk and E are its eigenfunctions and ci', v.'.ilues, respectively. The reference function cp is also a trivial ei: r..'t Lion of H with e iLi.'lu e zero. All the other eigenfunctions [(k] are restricted to be in the subspace of P. From a simple theorem about outer projections, it is easy to see that Ek Ek in order, where [Ek) are the eigenvalues of H for the eigenfunctions Yk [15(PTX)]. Let us now introduce the complex variable, and subtract g from both sides of (2.18) to give ( H)k = (e Ek (2.19) Using the fact that PY = Yk and defining the "reduced resolvent," T = ( )1P (2.20) it follows that (2.19) becomes T k = ( Ek)'lk (2.21) demonstrating that T and H have the same eigenfunctions. Therefore, the eigenvalue problem (2.21) is completely equivalent to (2.18), but T also contains the complex variable 6 which adds to the flexibility of the problem. It can be easily shown that T = TP = PT = PTP (2.22) and from (2.20), one obtains the useful relationship P(& H)T = P (2.23) which constitutes a connecting link with the eigenvalue problem (2.1). Let us now define a "trial" wavefunction TY which is dependent on the variable & by Y = cp + T He (2.24) Since T H8cp is in the orthogonal complement to cp Ye is composed of the reference function and a term from the complementary space. With the definition (2.24) we also have the "intermediate" normalization condition satisfied, that is (po IY = The intermediate normalization is a convenient choice to make since it is applicable to both the discrete and continuous parts of the spectrum. As long as the normalization integral (T Yl ) exists, Y is said to belong to the discrete part. Using (2.23) it is seen that the trial wavefunction satisfies the algebraic identity P(H &)Ng = 0 (2.26) for all the values of 8. Also, by using the fact that &+ P = 1, we have (H 8)Yg = (p 0 ) (2.27) where 8 is defined as o o = Expression (2.28) is the operator form of the "bracketing function" that is equivalent to (2.11). Hence TY satisfies an inhomogeneous equation which becomes identical to the eigenfunction in (2.1) when & = P = E . o o The trial function (2.24) and the expression (2.28) are essential to the following treatment of BrillouinWigner and RayleighSclirUdin;'i r pertur bation theory. 11.3. BrillouinWigner Perturbation Theory [16] Assume that the Hamiltonian for the system can be separated such that H = H + V, (2.29) where V is a perturbation which may or may not be small. hli,,n if the 16 previously defined reference function cp is assumed to be the unperturbed solution with T the eigenfunction to the entire Hamiltonian, one may arbitrarily define Y = W(p (2.30) where W, often termed the "wave operator," takes op into Y. Again, as throughout this paper, we shall require the intermediate normalization (poI>) = 1. (2.31) By multiplying the eigenvalue equation (2.1) by cp (the complex conjugate of cp ) and integrating we have E = (oI H 7) = (cpjo Ho+ ) = E + (cpjvo j), (2.32) giving the en,.rgy difference between the unperturbed result and the exact solution as an expectation value over cp of the quantity VW. As yet we still have no realization for W; but, by appealing to the previous defini tions (2.24) and (2.28) and using the relationship H Co = E 'p we obtain Y = (1 + T V)%o (2.33a) and S= ( V + VT V cp). (2.33b) For the case of 8 = o = E we have complete agreement with (2.32) for W = 1 + TE V, which defines the primary formulas of perturbation theory. In all perturbation treatments, we want to have expressions dependent on H and cp o. Thus it is convenient to define the reduced resolvent T o O associated with H in an analogous manner to (2.20), i.e., T = (& PH P)1 P. (2.34) O o 17 T can be related to T by means of the closed form of the identity (2.12), (A B)1 = A1 + AB(A B)1, (2.35) now pertaining to operators rather than matrices. With the separation (2.29) and using (2.35), we find that T and T are related by T = T + T VT, (2.36) o o allowing (2.32) to be written 8= E + ( IV + VT V + VT VTIp >. (2.37) 0 0 0 0 0 0 By repeated use of (2.36), an infinite geometric expansion for Tg in the quantity T V may be obtained, 0 T = E (T V)kT, (2.38) 0 0 k=0 leading to S= To + E (TV) o '(2.39a k=l and & E + (CPoV[ E (T V)k ]cp). (2.39b) 0 0 k=O 0 To further illustrate the connection with the more standard inhomogencous equation approach to BrillouincT.1n:r (BW) perturbation theory, we can define k = (V)kcp (2.40a) and S o I V I Pk) ; (2.40b) then once & = P = E (2.39) becomes o YE =o 0 + E Pk (2.41) k=l From the recursion formula, k1 = (ToV)k (2.42) and (2.23), we have the set of inhomogeneous equations (E Ho)Pkl = VPk k+1o. (2.43) Consequently, as long as it is possible to solve the equations (2.43), one can construct all the ({ck) from the unperturbed solution cp and the succes sive evaluation of the kl i's. 11.4. RayleighSchrodinger Perturbation Theory [17] In most applications of perturbation theory, the RayleighSchrUdinger (RS) type has been used since it generally has better convergence proper ties than the BWU variety, as well as not yielding equations that are im plicit functions of the energy, 6. This latter simplification is a conse quence of the unperturbed :nc1rgy appearing in the RS reduced resolvent rather than the 8 as occurs in T Instead of expanding the fundamental resolvent T in terms of V as before, we can instead consider V' = V (& E) (2.44) o as the quantity to be used in the expansion. If so, from (2.20) and (2.29) we find that T = R + R V'T (2.45) o o with the RS reduced resolvent defined as R = (E PH P)1 P. (2.46) In analogy with the BW case, the resolvent and wave operator become T = E (R V')kR (2.47a) k=0 and W = 1 + (R V')kR V. (2.47b) k=0 The connection with the conventional inhomogeneous equations of RS theory is somewhat more complicated than in the BW case due to the remain der terms which arise in each order from the & = E in (2.44). However, from the definitions (2.30) and (2.32) with (2.47), it follows that T = 0 + E (R V')k (2.48a) k=l and k=O If the expressions (2.48) are arranged after powers of V in the normal perturbation series expansions, E = $ + E V (2.49a) E o k=l and E E0 + E 6 (2.49b) 0 k=l k then the lower order terms may be identified as S= R V$ (2.50) R (V o  32 = R (V 6 )RoV = R (V I )$ 2 o l o o o 1 1 SR (V E)R (V E6)R V E R 2V 3 o 0 1 0 0 2 0 o : = R (V I)2 2 R o o 1 2 2 o 1 and Cl = < ov o> E1 o V o E2= E3 0 2 ( 0 0 0(1)R 0v0) E4 = (1V3> < VRo(VE6)Ro(VE6)R0oVI : 2<( VR2V l o> Eventually the pattern emerges that the RS perturbation wavefunctions $k can be defined recursively as k k+ = R (V 61 )k E R 6o E (2.52) +1 o 1 ~k o kA A with the RS energies simply E1 = These are the usual working definitions of the RS quantities. Using the intermediate normalization condition, (2.52) is easily framed into the inhomogeneous equation form by means of the simple relation P(Eo H )R = R (Eo H )P = P (2.54) 0 0 0 0 0 0 that is obtained analogously to (2.23). 11.5. Inner and Outer Projections Another important technique which is often used in connection with perturbation theory and variational studies is the method of "inner" and "outer" projections. Since we shall have several occasions for their use, it will be worthwhile to briefly mention some of their properties. An outer projection of an operator A is defined as A = 8 A'f (2.55) for 6f a projector characteristic of the space spanned by the basis f E). For a basis of either finite or infinite order, af may be written explic itly f = )ff Iff)l(^, (2.56) which is easily seen to satisfy the relations 0' = 0' and 0'f = 0 In the event that the infinite basis is complete, then O' is simply the identity operator. For a moment let us define A as the Hamiltonian oper ator for the system and assume that \k = I f[)k (2.57) is an eigenfunction to H. Then from the eigenvalue relation HYk= Ek (2.58) we have IE) ((flfif)1 H fk Ek) = 0. (2.59) Since it is assumed that the basis if) is linearly independent, HfI k f T k (2.60) where the metric matrix /f= (fE ) was introduced. Therefore, the outer projected Hamiltonian H has an exactly soluble eigenvalue equation, which is identical to (2.1) if If) is complete and corresponds to the varia tional solution if the basis is finite. In the latter case, the set of ;.v'.'lu.. ([k} are guaranteed to be upper bounds to the exact results 22 in order, since diagonalization of Ilf insures that each Yk is orthogonal and noninteracting with respect to those of lower energy [18,19]. A Hamiltonian of the form H is convenient since it allows one to formulate quantum chemical problems in a consistent fashion even though an approxi mate solution may be involved. If F and G are two Hermitian operators which fulfill the condition that (XIFIX) > (XIGIX) (2.61) for all possible X in the common domain of F and G, then it can be said that F > G (2.62) Consequently, if F > 0, F is positive definite. Now let us require that the operator A be positive definite. If so, we can define the inner projection of A as A' = A f A. (2.63) Since 0 Or f 1, A' satisfies the operator inequality 0 A' A. (2.64) By making the transformation i > = A Jf) > (2.65a) and go>= A affI) (2.65b) we obtain two alternative forms for A', namely the Bazley projection [20], 23 A' = h) (1hI A1 h)1 I and the Aronszajn form [21], A' = Agg) (. AJg)1(glA. (2.66b) The inequality (2.64) can only be concluded for A > 0, but expressions (2.66) will still converge to A in the limit of a complete basis, consti tuting a reasonable approximation even though the operator of interest has no unique sign. One important example of the use of an inner projection is found in the even order terms of a perturbation series expansion. For instance, in the RS case for R < 0, from (2.46) and (2.51), o E2 = (olVRoVIo. (2.67) With Pl )= Iln), A = (R ), and (2.64), we have R' ; R giving E6 = (JoVR'V[% 2', (2.68) which gives an alternate derivation of the Hylleraas variational princi ple [22,23]. It can be similarly shown that as long as %,1l ,..., n are exactly known [24], S (2n (2.69) 2n 2n for any n, a result obtained by Scherr and Knight in a different way [25]. (See also [26].) CHAPTER III. PADE APPROXIMANTS In all applications of perturbation theory one is concerned with various series expansions for functions or energies which quite often converge slowly or not at all. In particular, in modern perturbation theory the restriction that the perturbation be small has been essentially eliminated, often leading to very serious convergence difficulties that require more powerful summation techniques in order to extract useful information. One possible approach to this type of problem that has been shown to be fruitful is the Pade approximant summation procedure [27]. Although all the convergence properties of Pade approximants are not yet known [28,29], in the cases where mathematical justification is avail able, the range of convergence for a Pade' approximant has been found to be vastly superior to that of an ordinary power series. In numerous other practical applications where a rigorous mathematical proof is lacking, the results seem to indicate that the range of convergence is actually greatly in excess of that for the situations in which it has been proved. In the following, the basic concept and definitions will be briefly presented including a convenient relationship with the inner projection technique which will subsequently be of use. III.1. Derivation and Definitions Consider a function f(z) which has a formal series expansion f(z) = E a z. (3.1) 1=0 The [N,M] Pade approximant to this series is defined as the ratio of two 24 polynomials P(z) and Q(z) [N,M] = P(z)/Q(z), (3.2) which are respectively of degree M and N and coincide with the series expansion (3.1) through the (M + N)th power. By requiring the latter prop erty, the coefficients of P(z) and Q(z) are determined by the condition that f(z)Q(z) P(z) = (zm + n+ 1) with O(m+ n To prove that another ratio + 1) indicating zero up to terms of order greater than M + N. the ratio (3.2) gives a unique [N,M] approximant assume that (call it P'/Q') is possible, then f(z)Q(z) P(z) = O(zm + n+ 1), f(z)Q'(z) P'(z) = O(zn + m+ 1). (3.4a) (3.4b) Left multiplying (3.4a) by Q' and (3.4b) by Q, the first terms may be eliminated to give P(z)Q'(z) Q (z)P'(z) = O(zm + n + 1). (3.5) Since PQ' and QP' cannot be of higher order than M + N, to that order we have P/Q = P'/Q' (3.6) which establishes the uniqueness of the Pade"approximant [30]. If we write k P(z) = bk z k=O (3.7a) (3.3) N k Q(z) = E c k (3.7b) k=0 then be performing the multiplication in (3.3) and equating like powers of z, one gets the set of equations ac = b a c + ac = b1 oo o 1 1 ac + a c + ... + a c = b no n1 1 o n n (3.8) a c + a c + ... + mo ml 1 a ,c + ac + ... + mib1 o m 1 a c = b mn n m a M c = 0 mnH1 n a c+ + a c + ... + a c = 0. mn o mn1 1 min Since the last N homogeneous equations involve the N + 1 unknowns, {c.}, there is an infinity of solutions possible; but since the Pade approximant is only determined to within a constant factor, by arbitrarily choosing c = 1, the remaining (c.} can be obtained. With these c.'s the [b.) are determined from the first N + 1 equations of (3.5) and thus Q(z) and P(z) are specified. The explicit Pade solution to (3.3) is a amnr* 1 mal2 a m m a. z3 =nn 3=n . mn+32 a min3 n1 z nili a i a mrn3 m j j=n1 n+1 Z * am.I1 mi1 rin 1 subject to the convention that a. = 0 when j < 0, and any sums which have the initial element of higher index than the final element are excluded. Another form of (3.6) that is particularly convenient for computation is the inner projection form [15(PTX ),31,32]. (See also III.2.) If we define a = (a.a ..a ) i i i+N1 A. [a i ...a ] 1 i il i+N1 (3.10) (3.11) then for M = N + j, (3.8) can be written [33] [N,Nij] = Za + z l A j+l zAj+2 la j1 (3.12 where the poles are given by the solutions of A\ j+ zA\ .j 0. The expression (3.12) could be shown to be equivalent to (3.9) by [N,M] = (3.9) m E a.zJ j=0 I amn+l1 a m n+2 a m n z standard determinant manipulations, but the equality can be established more efficiently by expanding the inverse in (3.12) by (2.12); then Ij 2N1 1 [N,NFj] = a z + zj+1 a aiA (zA2 A . =0 "higher order terms" (3.13) + "higher order terms" (3.13) If we write 1 Aj+l bj+1 b j+2 bj+N where each b. is a row matrix of N of the inverse (3.14) elements, then from the properties bk a = 6k (3.15) With the notation Ik meaning a column matrix with the kth element unity and the others zero, we can establish the relations 1 A a = 1 1 k N ji1 j+k k (3.16a) A j+2k j+k+1 1 from which (A 2 A ) Ia1 = j+2 j+1 j+l j+., I I1( Aj01 j+ 2 j+ )+=1 1 and (3.16b) (3.17) (3.18) From (3.18), (3.13) finally becomes 2Nj [N,Nj] = E a z + "higher order terms" (3.19) =0 By the uniqueness theorem, it necessarily follows that (3.12) is equiva lent to (3.9) [33]. 111.2. Connection with Moment Expansions and Inner Projections [33] Let us consider two operators R and A related by 1 R(z) = (1 zA(z)) (3.20) and the expectation value of R with respect to a normalized function *, f(z) = (*IR(z) I). (3.21) If R(z) is expanded by (2.35), it follows that f(z) = E z (flA%). (3.22) 2=0 Defining a = (fIJAA I), a, is the Ith moment of A and (3.22) is the moment expansion of f(z). For any basis h ), an inner projected approximation to f(z) can be given in the Bazley [20] form by f(z) (f h) ( < 1 zA h)>1 (allJr> (3.23) Since the basis Ih) is arbitrary, we are at liberty to choose the functions  ) = AA24...) (3.24) from which it is found that f(z) a [A zA Il 0 (3.25) The result (3.25) can be identified from (3.12) as the [N,N1] Pade' approximant. Furthermore, since the expansion of (3.20) can also be written as 1 R(z) = 1 + zA(1 zA)1, (3.26) it is possible to give another approximation f(z) ao + zat[ zA12 ]1a1 (3.27) which is seen to be the [N,N] approximant. Similarly, the higher Pade' approximants, f(z) [N,N+j]'= E az + z j+(A+ zA ) j+ (3.28) l a +_1 j+02 j+ are derived. In the event that R is positive definite, from the inner projection property (2.64) it can be further concluded that [N,N] 2 f(z) [N,N1]. [33] (3.29) In this case the moment expansion is a series of Stieltjes and many addi tional conclusions may be reached. For other operators that are not of a definite sign, one can usually only treat (3.28) as an approximation to f(z). However, since in the subsequent discussion, the [N,N1] Pade' approximant will be identified as a partitioned eigenvalue problem, in this instance, it necessarily gives an upper bound for any finite basis. CHAPTER IV. THE REDUCED PARTITIONING PROCEDURE By a consideration of perturbation theory within the configuration interaction (CI) domain, we shall develop a procedure that allows an adequate representation for a CI wavefunction with only a small number of perturbation corrections to an unperturbed reference function. This pro cess, termed the reduced partitioning procedure, leads to several advan tages over the more standard approaches, one of which is found in the treatment of the full CI problem. The theory as formulated incorporates variational principles with perturbation methods to yield a scheme that has known convergence properties and one which is also very convenient from the computational viewpoint. In connection with the nonlinear summation techniques invoked, a modification of ordinary perturbation theory is obtained from which an alternate set of perturbation corrections to the energy and the wavefunction are defined. IV.1. NonLinear Summation of Perturbation Expansions In order to approximately solve the Schrb'dinger equation, HY = EY, (4.1) via perturbation theory, one usually assumes a separation of the Hamilton ian operator H = Ho + XV, (4.2) with H chosen to have an eigenfunction 0 and its associated eigenvalue o O 31 So, i.e., H $ = & 4 (4.3) 0 0 0 0 Within the RayleighSchrodinger framework, it is then assumed that both Y and E can be expanded in terms of an "order" parameter X, Y= o +U1 + 2 + ... 2 (4.4a) and E = 0o + XE1 + X2 + .... (4.4b) By substituting (4.4) into (4.1) and equating the various "orders" of X, one obtains the (Y.3 as solutions of the inhomogeneous equations n2 (e Ho)@ = (V E1)n E (4.5) o o n 1 n with the perturbation energies [(k) defined as Ek+l = Recalling the definitions I = I0 ) ( 0)j1(@I, P = 1 &, (4.7) and 1 R = (% PH P) p, (4.8) 0 0 then using R 0 = 0, (2.54), and the intermediate normalization ePk = k0 (4.9) (4.5) may be conveniently written as n2 = R (V E1 ) n Z R 6 oEn (4.10) If the expansions (4.4) are to yield a valid solution, they must con verge to the exact eigenfunction and eigenvalue of the Hamiltonian, H; but this may not be fulfilled for many choices of the splitting (4.2). Moreover, it is seldom possible to obtain solutions to (4.10) without re sorting to some approximate method, a complication which tends to further obscure the validity of the conventional expansions. To increase the flexibility of the problem, we prefer to consider the more general finite expansions (m) = + + Y 2, + + Xml' + X m(4.11a) o 1 22 m m and (m) = + XE + 2E + ... + + E1' (4.lib) for ) = ) (4.12) k k k The factors (c(m)), which may be dependent on the truncation order m, are at our disposal. From the definitions (4.6) and (4.12), the modified perturbation energies are seen to be Ek+1 = V ol ) = ckk+l. (4.13) In the event that the (c(m)) for all i and m are chosen to be unity, standard perturbation theory is regained leading to 1(m) = Y and &(m) = E at infinite order. In other cases, though, we have an additional facility that permits us to insure that (4.11) constitutes a valid expansion that 34 will allow *(m) and &(m) to approach and E as m increases even though the ordinary linear perturbation series shows no convergence. The BrillouinWigner quantities with 1 T = (e PH P)1 P, (4.14) are defined as k+= T )Vk (4.15) and ek+1 (= PIVIPk)' (4.16) where 8 = (9o1V + VT(F)Vlo)>. (4.17) With the previous modification, we similarly obtain = a(m) (4.18) and = 0jolV ) = akm) k+1 (4.19) Besides the conventional choice of c. or a. as unity for all i, we have the option of determining the coefficients from the variational prin ciple, or more generally the LippmannSchwinger extremum principle [34], where the functions (y.5 or (cpi) are chosen as the basis vectors. The variational principle has the important advantage of assuring that the modified energy expansion (4.4b) gives a converging upper bound in any order. Another possibility is to determine the coefficients by a Pade' summation of the perturbation energies given by.(4.6) or (4.16). This 35 last alternative can yield numerous different solutions of varying degrees of desirability. In particular, the BW energy expansion, (4.17), for a positive definite perturbation and for & less than the first excited un perturbed state, is a series of Stieltjes. From this it may be concluded that determining the a(m) from the [N,N] or [N,N1] approximants results in the resultant t(m) for m 1 yielding respectively upper and lower bounds to E [33]. (See III.2.) This last feature can also be re lated to the LippmannSchwinger principle [35]. IV.2. A Model Hamiltonian for a Finite Basis As it is our intention to study energies and other molecular proper ties as well as is possible within the nparticle basis if) we are necessarily limited to results that could be obtained from the full CI solution. As has been discussed, this in itself is a worthy objective sincee essentially all of molecular quantum mechanics makes use of some =sis set, and,as such, the whole field can rigorously do no better than a full CI within the selected basis. Moreover, in the vast majority of practical applications of perturba tion theory, especially if higher order corrections than p, are desired, the functions defined by (4.10) or (4.15) must be calculated in an approxi mate manner, usually by introducing a basis. Once the perturbation quanti ties are so approximated, another question arises, namely will the proper ties of these approximate solutions be those of the exact ones, and further, will the approximate ones converge to the exact ones as the basis tends to completeness. The basis [ff)= If f ...f ) in general contains all the configura tions of the correct symmetry wnich can be constructed from a suitably chosen Slater orbital or some other type of basis set. In order to suitably dispense with these inconveniences from the on set, we shall explicitly introduce the orthonormal basis Iff)by defining a model Hamiltonian as the outer projection of the correct Hamiltonian for the system, then C = fHf = jifF) (ffHff)(ff (4.20) for Of = jf) From the splitting (4.2), we also have 3C = 1H Hf (4.22) o fof (4.22) and ff= V&' (4.23) If we now partition the basis IE) into a reference function 0c and n ele ments required to be in the orthogonal complement to cpo, If)= cp0o,h *. (4.24) With this partitioning, a representation for the projector of the orthogo nal complement to CP is P = 1 o o><(pol = Ih> A useful relationship between the functions Ih) and an outer projected operator nf = f'fff is that ( *The partitioning need not be into one function and the remaining n, but may also involve a linear transformation among the elements, Iff). The subsequent discussion and particularly (4.26) will still follow. 37 By using the result of (11.5), it immediately follows that the model Hamiltonian (4.20) is exactly soluble, i.e., EY = Ei 1 (4.27) for Sfi i = ( o1 C2"'Tn), (4.28) with T the matrix of eigenvectors to the Hamiltonian matrix defined by If). In addition, (4.22) defines anX which always has a spectrum of exact unperturbed solutions whether or not H itself has a set of eigen functions. This last feature eliminates the usual perturbation theory tyranny of limiting one to a very few possible forms for H since 3C can be used for any form of unperturbed Hamiltonian which appears to be desir able. In the event that one actually has an H that yields a spectrum of eigenfunctions, Hok 0 k k0 (4.29) then by choosing Iff)= 1[e o...9 ), it follows from the definitions that Koek = k (4.30) also. Thus, the quantity KX can be viewed as another generalization of the ordinary theory. Relative to the finite basis set and the XC of (4.22), the previous expressions for the resolvents R and T become exactly [24] R = Ih)<(hi OC oj)1(l = h>R <0(j (4.31a) To = lh><)(lh osi h)'1( = Ah)To <(hj, (4.31b) where the relationships P lh) = [b>)and(4.26) were used. If the functions 38 SIh) are eigenfunctions of X then the inverse matrix is diagonal giving the more customary forms, n Ihk)(hkl R = E (4.32a) o k=l o and n Ihk)(hkl T = Z 0 (4.32b) o k=l E Then, subject to the intermediate normalization, the RS expressions are n = h)n (4.33a) n2 a = R [(IhV El6 jh)_ z E n ] (4.33b) and En = (oVh>) n; (4.33c) and for the BW, Cn = >1n, (4.34a) n = To(IhIVI h)> nl (4.34b) and Sn = (o I VI h)n (4.34c) These expressions (4.33) and (4.34) which follow from the model Hamilton ian (4.20) are identical, of course, to those obtained from the Hylleraas, Scherr, and Knight (HSK) method which yields approximate perturbation quan tities for the exact Hamiltonian H = H + V [22,24,25]. That is, instead 0 of solving the problem (4.1) approximately, we are defining a Hamiltonian C = KC +7/ whose eigenfunctions and eigenvalues can be exactly determined. o 39 Our results, however, are still slightly more general, since our 3C con o tains the HSK H as a special case. From (4.33) and (4.34) the elements occurring in perturbation theory can be precisely calculated, and the coefficients of the general expan sions (4.11) and (4.18) can be chosen to guarantee convergence to the ex act result for 3, ensuring that the orthogonality properties of the eigen function to an Hermitian operator are obeyed. Furthermore, as If ) be comes complete, the solutions of the model Hamiltonian must approach the correct eigenfunctions and eigenvalues of the problem, and by the varia tional theorem always from above. Similarly, by a selective choice of the coefficients in (4.11) and (4.18), the energy to any order of solution can also be made an upper bound  and sometimes a lower bound  which, con trary to much of modern perturbation theory, allows one to always know the relation between the computed result and experiment. IV.3. The Fundamental Equations for the Reduction Process In the previous section we defined a perfectly general K as a multi dimensional outer projection. We shall have occasion to use a form simi lar to this in the discussion of excited states, but for the present, another form arising from an additional outer projection relative to co' V= 'o o = ocp (cpj, (4.35) is found to be an especially convenient choice. Obviously, this V0 sat isfies (4.3) as well as allowing us to introduce any approximate wave function as the reference function in a perturbation formulation. From the partitioning (4.24), the definition (4.22), and *This particular form for W has been previously discussed by Musher [36] and Epstein and Karl [37]. 0 40 P I> = Ih), (4.36) it follows that Jcp)(o I fo f Jf)(f oo0)(p = )e ol (4.37) for e the expectation value of the arbitrary reference function p With this o we have the definition V = 3C % (4.38) The disadvantage of this particular V( is that it leads to divergent ex o pensions [36]. Thus in order to obtain meaningful results, its use neces sitates a nonlinear summation procedure to obtain the arbitrary coeffi cients in (4.12) or (4.18). Another feature of the ( of (4.35) is that it only has a single non o vanishing eigenvalue. This results in the R of (4.8) and T of (4.14) o o assuming the simple forms Ro = P/& ; T = P/&. (4.39) Restricting ourselves for the present to the simpler BW expressions, in the following, we shall deduce a reduction procedure by appealing to the properties of a moment series expansion and the use of Paddf approxi mants. The final closed expression can be identified as an inner projec tion and, as such, guarantees an upper bound in all orders. From the de fining equations of BW perturbation theory, (2.37) and (2.40), we have S= ( and k (TV)kl1 = (ToV) ko). (4.41) By using (4.38) and (4.39), with PY0 = 0, we obtain S= + + I/j( pf CPeCcp) + l/ 2(co JC(P3C)2 o >+ ... (4.42) and qk = l/&k(P3C)kPo>). (4.43) Then, with the definitions S= (PC)klo) =lh)ihjHJIh)kl IhHIcp0> (4.44) kI S h) H lhh b k > 1, and k = oHk) = k hh Ib, (4.45) the 1/8 moment expansion (4.42) can be written as P = + + // + l/ 2 a + ... (4.46) o 8 o 8 1 where the previously defined BW energies are k = 0k2/ Consequently, given the Hamiltonian matrix in the complementary space, Hhh' to con struct the quantities (4.44) or (4.45), one only needs to perform various matrix multiplications. In principle it should be possible to solve (4.46) as an implicit function of e, and if the convergence is sufficiently rapid, obtain a good answer for the eigenvalue. However, this is impossible for two important reasons. First, it is expected that the series (4.46) should be strongly divergent. For example, for the series to converge the eigenvalues {Ci} of Hhh should be such that 1 < i./8 s +1 for each C, [38]. These condi tions are unlikely to be satisfied for molecular problems. A second com plication is that for any truncation of (4.46) there is necessarily a 42 dependence on the energy zeropoint. However, since Padd approximants are known to have better convergence and existence properties than a linear series, by using such a summation technique to determine the coefficients in the elements of the modified energy expansion given by (4.19), it may be possible to improve the situation. In particular, by applying the [M,M1] Pade"approximant for M N, (4.46) can be written as e(m) = g + e l/(m) [M,M1], (4.47) where [M,M1] = (E 1/ (m) E (4.48) and S= [.i i+1...om i = 0,1,2,...,m (4.49) Ek= k+l ... k+m1] k = 0,1. (4.50) With some slight rearrangement we have o(m) = + C (m) E E )I (4.51) 0 0 0t 0 This is the fundamental formula for the "reduction" process, but as yet it is not obvious if the two previous objections are eliminated. Con sequently, let us now consider the functions 1i) defined in (4.44). If we augment the first m of these functions by cp and then assume that (cpo j) is a reasonable basis for an approximate solution of (4.1), by using the fact that (cp ok = 6ok, we obtain the partitioned form for the secular equation equivalent to (4.1) as (m) = + H (g(m)f 1 H (4.52) 0 ocp cp p ( po where the quantities are defined as 43 A = (m)l(m), (4.53a) S = (m)Hl(m), (4.53b) and &(m) is the eigenvalue obtained as a function of the M basis functions 1(m)0). From (4.44) and (4.45) it also follows that pcp = 1 02 ... m (4.54) G2 3 m +m1 ... C2ml and A = I ... mI (4.55) fx o0 rm1 1 02 m1 .. 2m2 , which with m)= (m)) allows the identification with (4.51). In addi tion, the intermediately normalized Mth order wavefunctions are given by 1r(m) = + (m)& (m)o 1) 1 (4.56) Therefore, the summation of (4.46) by means of the [M,M1] Pade' approxi mant is identical to solving the M1411 x M+1 secular equation equivalent to (4.52). As such, besides elucidating the nature of the improved result given by the Pade summation, (m) for any m is guaranteed to be an upper bound to the full solution; and the result obtained from a variational calculation must be independent of the energy zeropoint. In the event that M = N, we have simply carried out a linear transformation of the original problem which is necessarily equivalent to the full CI solution. The expressions (4.51) and (4.56) are also seen to be consistent with the definitions (4.18) and (4.19) where the arbitrary coefficients are obtained via the variational principle. The degree of success of the re duced partitioning procedure, however, depends on the [M,M1] Pade sum approximating the [N,N1] result even though M << N, since then ,(m) should be very close to 8(n) = E, and hopefully, to a desired numerical accuracy. If so, this should effectively negate the necessity of solving enormous secular equations that would arise in the conventional approach, as well as leading to tractable solutions for further use. The equivalence of the [M,M1] Pade approximant and the variational solution in the basis Iop Z( ) could have been concluded in a slightly different fashion from the deduction of (111.2). This result shows that a moment expansion of which (4.46) is an example can be related to an inner projection which in turn is identified as an [M,M1] Pade'approxi mant. The fact that it is an upper bound follows from T being a nega tive definite operator. Having given the basic equations within the framework of BW theory, it may be asked what happens in the RS case. Recalling the recursion expressions (4.33) and (4.34), and by using (4.26), (4.36), (4.38), and (4.39), we have k2 k = lh)k '; k /E Chh k 1 k (4.57) o 0=1 and 'k = (>hk ; lk = hh'okI (4.58) where (4.58) is slightly modified from (4.34) to be consistent with (4.44). From these two relations, it can be seen that. 45 VW = I/ (4.59) 0 02 = 1/ e 2 Mhh 1 1/ 2 12 o 0 W3 = 1/eg3 1hhF2 E2/ A P o 0 This can be written A = B T (4.60) for A = ( 2 ... ), = (1 2 m); and, where it is also ob served that T is of upper triangular form. The latter fact ensures that for any order m, the variational solution relative to either basis set is the same; that is, =(m) = + I (m) ) (m) o (Im) a (m) (4.61) and e(m) = (m) iC(m) (4.62) Although the variational solutions in the BW and RS bases are equiv alent, a slightly different result could have been obtained in the latter case if the coefficients (c m)) were determined by applying a [M,M1] Pade approximant to the series (4.4b). This result is usually only slight ly different from the variational solution [24]. IV.4. The Method of Steepest Descent and the Geometric Sumrule In ordinary perturbation theory, the form of the inhomogeneous equa tions for the perturbation functions is dictated by the requirement that (4.4) or (4.40) is convergence. In the case of a divergence expansion, it is not necessarily obvious why the set of perturbation functions should still be one which rapidly exhausts the space of interest, as is neces sary for the reduction process to be effective. To attempt some justifi * cation of this point, we appeal to the method of "steepestdescent" as employed by McWeeny [40] in connection with SCF studies. Given an arbitray trial function, cp = cp + 8cp, where it is assumed that (cpo 6p) = 0 and eo =0 (cpojHjo), the energy increment due to the first order variation of the wavefunction is ( 68 = (CP)+ &P >4.63) If 1 h) constitutes a basis in the orthogonal complement to cp we can assume an expansion for 85p of the form &p) = Ilh>) (4.64) for e1 the expansion coefficients. If we neglect the second order term, (69cjH 6&p), and ask for the direction of the vector in the linear space corresponding to the maximum possible change in 8g, we obtain 6(68) = 0 = 8a 1 + [(lhH Jlh) 68l] (4.65) or, that t 1 is proportional to (IhJHcp ) = b as defined previously. Con sequently, the basic functional form for the first order correction should be 1= lh)lb (4.66) The optimum amount of cp1 to mix with cp o, that is the step size T, can be *Although philosophically similar, the terminology "steepest descent" usually pertains to a different application originally discussed by Rie mann and Debye [39]. determined by including the ignored second order term and requiring that S=0 0 (4.67) for P(T) = ( Then we have the result that = ())Go 1 ) 10 (4.69) where (4.66) and the definitions (4.45) were used. This result gives the first order equations obtained via the steepest descent procedure as g(1) = + o (e(1) 0 1) (4.70) and (1 = o+ 1 (1~o 1 (4.71) These two equations can be identified as those corresponding to (4.51) and (4.56) for n = 1, or the [1,0] Padedapproximant solutions. We can similarly justify the higher order expressions by considering the function (4.71) as a new reference function and repeating the proce dure where now (cpo 6*) = 0, but (4(1) 6>) # 0. If so, by again excluding the second order term, we have 6 1 )H( )1) (1) H (1) (1) 6[<6e[ >) + (<6 5 ) + ( By assuming 6* = Ih) 2 we find that 48 C 2 ~ hhb + pb (4.73) where the constant p indicates a mixing of some amount of b Thus, the next function which should be employed to augment cp1 is exactly the c2 of (4.44). By a continuation of this argument, the remaining (p's can be similarly justified, thereby providing an alternative explanation from a source external to perturbation theory for the special forms of (4.44). The optimum step size subject to po' ,cp and cp2 is obtained from a form (2) = po + 1 + I2, where l and T2 are variationally determined. This, of course, is the [2,1] Pad~eapproximant of (4.51) and (4.56). Another approximation could have been obtained by assuming the ) of (4.71) to be a new reference function andaskingfor the steepest change in the first order energy relative to this unperturbed solution. This type of procedure has been employed in a RS framework by Dalgarno and Stewart [41] and Hirschfelder [42,43]. For the BW case, compare also Goldhammer and Feenberg [44] and Young, et al. [45]. Besides being directly derivable from the steepest descent procedure, the first order solutions (4.70) and (4.71) have some other properties which make them particularly significant. For example, their use corre sponds to the application of a geometric sum rule to a series, in this case (4.46), and hence is often referred to as the "geometric" approxima tion. Despite the fact that such an approximation is seldom warranted by the actual terms in the series, it has been used with a great deal of suc cess in a variety of applications [46,47,48,49]. This approximation  recognized as the [1,0] approximant  has been previously justified on a variational basis [50], as in the present case, where it rigorously corre sponds to a variational solution corrected through the first order in the wavefunction. 49 As should be expected from the connection with the method of steep est descent, this first order solution gives a substantial improvement over the unperturbed result, yielding, in the caseswe have studied, about sixty to seventy per cent of the possible energy improvement that may be gained from the given basis set. Furthermore, the geometric solution also has an important simplifying feature when the reference function is chosen to be the two most common types, namely the HartreeFockRoothaan (HFR) or the CI variety. Let us consider the first case where Co is chosen to be an HFR wavefunction and assume the remainder of the configu rations are generated by the substitution of a "virtual" orbital. If we partition h) such that the single, double, etc. excitations are explic itly introduced, i) = b (1) (2)... (n), (4.74) then from Brillouin's theorem it follows that 0o = (PoHlI0 (2)h (2) IHjcp (4.75) and 01 = (y IHl h(2) )h(2) H h (2)) (2)h. (4.76) Consequently, the [1,0] solutions (4.70) and (4.71), relative to an HFR function, depend solely on the double excitations. The CI reference function will be discussed in the next chapter where excited states are considered. CHAPTER V. EXCITED STATES The subject of excited states is an important one in quantum chem istry, since it is an area where a great deal of information, which is not readily accessible to experiment, can be obtained from theoretical consideration. Compared to ground states, however, the difficulties associated with an adequate determination of excited states are propor tionately compounded. In particular, it is expected that the higher ex citations that one usually neglects in a CI wavefunction will be of more significance for excited states, from which it would appear that a reduc tion process should be of even more importance. In the following, a supermatrix formulation for excited states, which is a generalization of the previous theory, will be presented. From a consideration of these equations, a modification is made which allows a selective treatment of the specific state of interest. This last result will also suggest a con ceptually appealing approximate treatment. As in the HartreeFock case, the first order solution for excited states is found to have a computa tionally convenient property. V.1. The SuperMatrix Formulation In order to progress to excited states, it is necessary to consider a multidimensional reference space f) = XXI'..X p), which has the characteristic projector The multidimensional partitioning technique has been employed in lower bound studies by Choi [51] and Wilson [52], where the reference space consisted of a degenerate set of functions. Y = y . X Within the space spanned by If), for If) = lj h), &f is the identity operator, thus P = & = Ih)<(IhI. Consider an eigenvalue equation, 10 = Y.E. for some state Y. with eigenvalue E.. Using the property that x= O< = C8,f = 'YCf = efHf, (5.4) can be rewritten as C(0 + P)Y. = Y.E. . From (5.6) the two equations, SY. Ti x i E XPY i 51 (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7a) and P xC i. = PE.Y. P3CPY., (5.7b) are obtained by left multiplication of 0' and P, respectively. Solving x (5.7b) for PT., it follows that PYi =TE 3X. i (5.8) where TEi has the usual definition TE = (E. PXCP)1 P. (5.9) E i By substituting the expression (5.8) for PYi., (5.7a) becomes e& ( + XKTEI )eXYi = (5.10) or that Q Y. is an eigenfunction to the operator 8' (K + 3CTE 3C)8 with Xl X i X eigenvalue E.. (See Lowdin [15(PTIV)] .) Thus, (5.10) is exactly equiva lent to (5.4), but a consideration of (5.10) leads us to a perturbation treatment. With the definition (5.1), the fact that Y. = Iff)C and the par titioning (5.2), (5.10) in matrix form is yX jX + XT EiX he ff) C = ><)(Vif)C.E,. (5.11) For [j) linearly independent and C= ( Xi), this requires that Chi XX C = E. (5.12) This gives an alternative operator derivation of (2.10), which was pre viously obtained from a direct partitioning of the eigenvalue equation. Obviously, the essential difference between the basic equation for the multidimensional case, (5.12), and the earlier example for p = 0 is that. now we cannot remove C to get a direct expression for the energy E.. Xi Instead, we must solve a secular equation for an effective Hamiltonian. This consequence ensures that the resultant multiple state solutions will satisfy the requisite orthogonality and noninteracting properties. Since the expression (5.12) is an implicit function of E., in practice we must resort to the more general form XH (P) t (P) = X (8)f (5.13) xX\ x X where for eachchoice of 8, an H (8) may be constructed, from which XX diagonalization yields p + 1 roots, ({ ...p ). The eigenvalues (E.i are given when P = F. = E. for some i. With the principal definition, a () = < X + XT r3 >, (5.14) we can proceed as previously by expanding (5.14) via the relation T = T + T XT, to obtain o o XX (8) = IX + (IKCT 0 + CT 0XT + ..."'). (5.15) With the definition I k(&) = (I3C(To0C)k+l 4), k 2 0, (5.16) (5.15) can be written as an expansion in terms of matrices xx () = xx () + 1() + ( (+ ) + 2(8) + .... (5.17) Now let us assume the set y) are the solutions to an unperturbed pro blem, that is, C y) = ) to, (5.18) for t a diagonal matrix consisting of the unperturbed eigenvalues, 0o ..o 3. Then the expression (5.17) becomes the multidimensional o 1 p equivalent of the usual BW perturbation expansions which, by taking to a certain order and then solving (5.12), yields the perturbation correc tions for each of the p + 1 roots. This type of treatment for degenerate RS perturbation theory was originally given by Van Vleck [53]. (See also Hirschfelder [54].) For a general XC we could apply various treatments 0 to (5.17) to obtain series of nonlinear expressions for IXX but in the interest of brevi\y, we shall first specify a particular unperturbed Hamiltonian. Analogous to thi previous single partitioning example, we shall select for the unpert ibed Hamiltonian 3C C e 0 (5.19) which is easily seen Lo satisfy (5.18). Using the spectral expansion and (5.3) we have, just as before, T = P/ o Then from (5.16), !J.+1 = 1/ k+l (y IC(PPC)k+lIl), (5.20) (5.21) where the ([k) are imant to (5.17) we reduced, effective H:r () (m) XX e,ndent of &. By applying the [M,M1] Pade'approx in to order m the supermatrix expression for the 1 tonian matrix, (5.22) The supermatrices U and 14 are defined by 2 = ytt 12 m 'r' =71 2 + 2" am r U LA+1 lml "Al' AF2ml 'A2m1 A = 0,1,... a = 0,1,... and (5.23a) (5.23b) fk o + (efo a )i .o 0 0 0 1 0 Inspection of (5.22) indicates that the supermatrix product results in a p+ x p+1 matrix. A somewhat similar kind of treatment has been given in [15(PTXIV)]. To further illustrate the connection with the previous theory, we can define the perturbation corrections relative to each unperturbed state, (i) = k H kk i) = (PX)kiXi = j khhl Hxi) = 'l) h bi (5.24) which should be compared to (4.44). Also, we have the analogous sigma quantities i) I (i) b I k lb (5.25) k YXiIcPk i hh i and for i # j, i, j) = (x IJpP) = (xi KJ3C pi) 11h (5.26) which are generalizations of (4.45). The terms (i are required to describe the interactions arising from the different states ^). With these definitions, the elements of the k matrices are seen to be (o) (1) (p) C= k k "k ] (5.27) where the sigmas are grouped into column matrices, (0,i) S(i) k (p, i) As before, we can identify the basic reduction equation for multiple states, (5.22), as the partitional form of an eigenvalue equation, but this time relative to the basis jyp() () ...(p)>, for (i) (i) (i) (i)(5.29) = 1 9l 2 "'' m ) (5.29) When p = 0 and i = 0, the fundamental expression, (5.22), reduces to the single dimension case. V.2. The Selective Reduction Procedure The advantage of a reduction process based on (5.22) is that all states are treated equivalently, giving a set of p + 1 improved states at any order m. This is also a disadvantage, though, since in order to effect this result one must use (m + l)(p + 1) functions in the computa tion. In a typical CI calculation, even though we expect the order m to be low, p may be a rather large number. To avoid this difficulty, a imod ification of this method will be presented, which, requiring only m + p + i functions, permits a higher order treatment for one specific state. Although it is possible to obtain the desired result directly from a consideration of the supermatrix equations, for illustrative purposes it is convenient to approach the problem from a different perspective. Therefore, let us consider the possibility of obtaining a wavefunction for each excited state from a perturbation treatment of its corresponding unperturbed solution. Then, as in the single partitioning example, we assume a trial function of the form S= (1 + TP)Xi, (5.30) consisting of a single element in the reference space ^), and a term from the orthogonal complement, jIh). Contrary to a rigorous single parti tioning, ei cannot become the correct solution Y. when 8 = = E., as long as i contains any contributions from the other elements of jy). However, in the usual application of the theory, the set I ) will be eigenvectors of a relatively small CI problem and will be necessarily mutually orthogonal and noninteracting. The only mixing allowed is found in the Hamiltonian matrix elements involving the remaining (Xj} and the terms from the orthogonal complement. Consequently, it could be expected that the contribution to .i from the additional (xj} is small. In any event, we can obtain the approximate expression E. (Xil3j Si = (XiC+KTTC IXi (5.31) from which the perturbation functions (5.24) are obtained, and the mt order reduction equations analogous to (4.51) and (4.56) are derived; (m) (m) =o (i) (i) (i)1 (i) S+ 0 ( E 0 E (5.32a) 1 1 1 o o 1 o and (m) (m) + (i)i) (i) (5.32b) Xi + (SE E ) o. (5.32b) One way to retain the rigorous upper bound properties of the p + 1 approximate solutions, (5.32), is to construct the Hamiltonian and over lap matrices relative to ~Tol...Tp3 and to solve the secular problem. Then one regains the noninteracting property required by the separation theorem [18,19]. This variation is not very flexible, though, since the relationship between the [x and ([P i) is already fixed. A superior alternative can be deduced by proceeding somewhat differently. Within the space of interest, and subject to (5.19), if the expression(5.14) is writ ten in more detail, we have *The solution corresponding to Xo, the ground state, is still ensured to be an upper bound although it cannot quite converge to the best solution in the given space If)= I ,Ih) since some of the Xj are excluded. XX o oh ho ohq h oh hp khQ() "ho (5.33) Ph ho ph() hp 1 for Q() = (1,hh hh 1" From this exact expression, (5.32) is seen to correspond to a reduction process applied exclusively to the diagonal elements. Consequently, assuming the perturbation functions so obtained, ( ...) are a good set to use in a selective reduction treat ment, it follows that we can construct several reduced Hamiltonian ma trices, one for each p(i), similar to (5.33), by simply taking the off diagonal elements into account. By so doing, we get to order m S(i,m) +F(i) (i) (i) (i) M) + 1P (e) eM )MP (5.34) XX o o o 1 o where (i) = (o,i) (l,i) (p,i)], (5.35) Pk k k k and ti,) = (i,j) (i,j) .0(i1,j) (5.36) k k+ k+m1" This form corresponds to a partitioned eigenvalue problem in the basis IXpi)), thereby allowing complete flexibility in the determination of the variation solution to yield the best possible result. At the same time, the proper noninteracting properties are also guaranteed, ensuring upper bounds for all roots. Since one may construct p + 1 effective Ham iltonian matrices of the form (5.34), diagonalization of each one yields (p + 1) eigenvalues and eigenvectors. From the previous discussion of steepest descent, however, it follows that by choosing i as the particular state of interest, the i root should be selectively reduced for m < n. When m = n, (5.34) for any i becomes exactly equivalent to (5.33) giving p + 1 identical secular equations. Besides the rigorous solutions, re sults obtained from the approximate expressions (5.32) will also be found to be useful. V.3. The Geometric Formula for a Multi Dimensional Reference Space In a practical application of the multidimensional reduction pro cedure, one is interested in a CI problem in the space jfE)partitioned into two subspaces Is) and Itt). Typically, Is) is composed of a selec tion of n configurations, neglecting the remaining nt configurations which are grouped into Itt). The dimension of Is) is usually chosen to be much larger than the number of roots desired. In such a case, the first order solution, a type of geometric formula, has a convenient simplifying feature. Let us consider If> = Is t) (5.37) for jff)an orthonormal set of configurations. The equation we would like to solve is (5.4), or equivalently, HI (f = f E. (5.38) In practice, though, we usually must resort to solving a truncated prob lem within the chosen space, IsY. By so doing, from the definitions (4.20) and (5.37), we have (s<13s) = (sIHIs) = ss (5.39) Ss and SssD =Ds 0 (5.40) SS S S O for the o of (5.18) and the ns possible eigenvectors, )s = d o .. n 1] (5.41) s Since we are only interested in the first p + 1 solutions, let us define them as S= Is) i, i = 0,l,...pSns. (5.42) These wavefunctions will constitute a multidimensional unperturbed ref erence space to implement a perturbation solution of (5.38). Now we may partition Iff) such that Iff) = 1g,v, tt) = I,lh>, (5.43) where ) consists of the remaining ns (p + 1) functions, which may be obtained from the basis js). The I ) are assumed to be orthogonal to Iy), and an adequate realization would be the other eigenvectors of (5.41), which are necessarily noninteracting with the unperturbed solutions. The following is a consequence of this latter property. The first order solution to the supermatrix equation, (5.22), is (1) + 1 T( = t + it (e yi) 1i" (5.44) From the relation (5.27), the important components are the { (i'j)] de fined in (5.26) where k = 0 and k = 1. With the definition of K and by using the eigenfunction properties of the jt), it follows that 0(ij) = (iHIJtt)(ttHIxj> (5.45) and (ij) = ( for any i,j = 0,1,...,p. This has the convenient consequence that in a truncated CI calculation the first order improvement to the p + 1 lowest states can be obtained by ignoring the ns (p + 1) combinations, I), and solely introducing the excluded configurations, Itt), into the calcu lation. Since the selective reduction equations (5.34) and the approxi mate solutions (5.32) are subclasses of (5.22), this feature is also present for these methods. Although we employed the n (p + 1) remain s ing eigenvectors of (5.41) in this derivation, in the interest of going to higher orders, it may be mentioned that this is not required since only the orthogonality of the l ) to the unperturbed solutions, IY), is sufficient to ensure the noninteracting property. When one selects a set of configurations for a CI calculation, it is assumed that those remaining are relatively insignificant. By determining the first order improvement to the initial choice Is) by means of (5.45) and (5.46), this supposition can be placed on a more quantitative basis. Its assistance in choosing important configurations is also apparent. Furthermore, since this solution typically accounts for more than fifty per cent of the obtainable energy improvement, it seems that it would be a logical extension of any standard CI computation. CHAPTER VI. NUMERICAL APPLICATIONS OF THE REDUCED PARTITIONING PROCEDURE In order to undertake anumerical assessment of the reduced partition ing theory, we made some numerical applications within the configuration interaction framework. Since one of the objectives of the theory is to minimize the difficulties encountered in configuration selection, we were interested in limiting these initial computations to molecules where the full CI wavefunction for a basis of several functions would be easily tractable, since the full CI, being the best answer in the chosen basis, will provide the ultimate result to use in our comparisons. Practically, this limited us to the twoelectron systems H2 and HeH+, as the number of initial basis functions allowed by our computer program (18) would lead to a few thousand configurations of the proper symmetry for even LiH. The calculations that we have performed fall essentially into two categories: ground state studies relative to a single unperturbed solu tion,and a simultaneous treatment of several states by employing a multi dimensional reference manifold. The first of these will be illustrated with equilibrium studies of H2 relative to a simple Coulson molecular orbital wavefunction as the unperturbed state as well as with a Hartree FockRoothaan reference function. In addition, a complete potential curve for the lowest + state of HeH subject to a HFR unperturbed solution will be obtained. As an example of the multidimensional case, the selec tive reduction of the EC+ excited states for HeH will also be considered, including some numerical justification of the approximate treatment pre viously described. Although this discussion is selfcontained in the 62 63 sense that the objective is a theoretical solution, due to the consider able experimental interest in the reactions of metastable He [55,56], these calculations also may be of some interest in connection with this problem. VI.1. Summary of the Computational Procedure The computer program implemented for this problem couples the in gredients of a configuration interaction calculation with the perturba tion approach described in the previous chapters. The initial basis functions used are the complex form of the Slater type orbitals (STO's) located on each center in the molecule. The complex form ensures that each orbital is an angular momentum eigenstate and thus assists one in constructing nparticle functions of the proper symmetry. From the initial basis of STO's, a secondary basis consisting of an orthogonal set of orbitals is generated. In the calculations presented here, these are either the Schmidt orthogonalized combinations of the STO's or the HartreeFockRoothaan eigenvectors one obtains from solving a preliminary single determinant problem. By associating a or $ spin with each member of this secondary basis, a set of oneparticle pure spin orbitals is ob tained. Due to the symmetry of the STO's, the resultant spin orbitals are specified by the usual 0, n, 6, etc. classification. From the spin orbitals, nparticle antisymmetrized products are constructed, where the symmetry of the state E, 1, A, etc. is easily achieved from the spin orbi tal specifications. For homonuclear diatomics the gerade and ungerade symmetry for the orbitals is also taken into account. By combining these antis :nillltrized products into the proper combinations, spin symmetry is incorporated into the elements of the basis set, as is the plus and minus s:,mctry arising from reflection in the plane of the molecule for sigma states. A configuration.for a diatomic molecule is thus defined as a pure symmetry function in all respects, as this gives the minimum possi ble number that must be considered for the problem of interest. The molecular integrals are computed from a program written by H. H. Michels and altered by J. Kouba to give integrals accurate to at least eight figures. Additional routines from J. Kouba's CI program [57] have been modified by the author for inclusion into the completed program. When HartreeFock wavefunctions are required, they are obtained from a program written by H. H. Michels, but with the more accurate integral routines substituted. It may be worthwhile to briefly summarize the essential steps in the remainder of the program, which is used to effect the perturbation calcu lations. This section is arranged to calculate CI wavefunctions relative to a given set of configurations, and then make two subpartitionings in each single run. With each partitioned result constituting one or several unperturbed states, these solutions and the remaining elements of the orthogonal complement are used to construct the perturbation quantities by the theory previously described. In this respect, routines have been written which allow us to work either within the RS or BW framework. Since a variational solution relative to the perturbation basis is desired, this can be accomplished by generating the respective Hamiltonian and overlap matrices from which the eigenvalue equation may be solved. In the BW case, this is easily accomplished solely from the perturbation energies, since from the basic definitions, (2.40), it follows that (CplV 1 q) = ( pl V(To V) o) = (PpqlV 'F= p+ql (6.1a) and (plpq) = P~q (6.1b) 65 In the RS case, the overlap matrix elements must be calculated from the perturbation corrections (2.52), but from these overlap elements and the perturbation energies (2.53), the Hamiltonian matrix elements are efficiently constructed via a relationship derived by L'wdin [15(PTTX)] giving p q pVl) = EP l + kE E= E kl. (6.2) Within BW theory, an iterative procedure for obtaining the [M,M1] and [M,M] Pade' approximants by means of the solution of the implicit equation, 8 = f(s), has also been implemented. This routine incorporates the technique of NewtonRaphson iterations [15(PTI)], a second order pro cess, to increase the rate of convergence. For example, for the [M,M1] approximant with a single partitioning S= () = E + E+) To (6.3) 0 0 0 1 o from which one readily obtains the quantity f'(t) = ( E E E 1 (6.4) which is always negative. From a first order approximation (1) =f(() (6.5) the improved value, (o) (1) _(o)) f,(0(o) = (O + [((1) )/(1 f( )], (6.6) follows from an application of the NewtonRaphson formula to F(8) = P f(s) = 0. (6.7) Furthermore, the fact that f'(&) < 0 guarantees that for (o) >e, (1) * G) < ; but e which can be related to the variational principle [15(PTI)], must always be greater than 9, ensuring ccr'.verglce from above. This process is continued until two successive &'s differ by no more than 100. Since this method requires repeated inversions of (E 0 E 1) for the various values of 8, although it may be more desirable in some respects, it is a more time consuming process than the equivalent secular equation solution. Despite the particular approach used, one encounters computational problems with near linear dependency in the perturbation basis set. In the implicit function process, this complication is manifested in the inversion of the illconditioned matrix (EE o E), and, in the eigenvalue equation, by the removal of the overlap matrix. In the program, this possibility is investigated at each order of calculation by multiplying the matrix with its inverse, or checking the transformed overlap matrix. As the perturbation functions rapidly exhaust the space of interest, a resulting near linear dependency is indicative of having essentially achieved the desired solution, but in the process it is often of interest to gain another one or two figures of accuracy. Consequently, to obtain slightly higher orders of solution than would otherwise be possible, a multiprecision extension written by C. E. Reid [58] for the IBM 360/65 has been incorporated into the program. With this package, multi precision routines for the Pade' approximant analysis, including a matrix inversion subroutine, have been written. The use of these routines de pends on the assumption that the pertinent quantities need to be calcu lated only in double precision (16 figures). Then with these values, the remaining analysis involving the more sensitive steps is accomplished with four words (37 figures) of precision. This process can be trusted up to the point where the linear dependency once .ig:in manifests itself, or when the ultimate effect depends on differences beyond the range of the low precision of the initially calculated quantities. Either of these features becomes apparent in the course of the calculation. In order to treat multiple states with the selective reduction pro cedure, the program uses essentially the same computational framework as in the single partitioning case. To illustrate this, let us consider the secular equation approach which requires constructing the "reduced" Hamiltonian and overlap matrices, (i) XX X(i) H Ri) X= (6.8) H(i) (i) and A Ax (i) AR (i) x(i) (6.9) cPX The partitionings are defined as S = (6.10a) XX o (i) = (pi) W (i,0) (i,l).. (i,p), (6.10b) XT= 1 2 m HM(i) (i)jXlV(i)) = (i) 1 and A = (J~y = 1 (6.11a) XX X A(i= (il( ) = (6.11b) Xcp A(i) = (i) (i) (i) (6.11c) for the reference functions, I ), and the sigma terms defined in (4.49), (4.50), and (5.26). Since we have t from the unperturbed solution, the quantities needed are Hi i ,and A To obtain these matrices, it is con X~P cpcp venient to begin by calculating the approximate solution, (5.32), for each of the p + 1 states. As this is a single partitioning approach, this is achieved with the basic program structure. In particular, the quantities bi = and the perturbation functions () = lh)0(i) (6.13) are generated for each i. From these, one also produces the matrices (i) (i) iH and A which defines the secular equation for the approximate solution, (5.32). By solving this, the approximate energy, &, and wavefunction, '., are given as the lowest energy solution. In the pro cess of obtaining each of these approximate results, by simply storing the L Ib., H ) and A\ externally, all the basic information necessary for the rigorous multiple states solutions is available. The remaining elements required are the terms (o' ij) which arise from the interaction of the unperturbed states, and are thus unique to the multidimensional case. These present no problem, though, since from the definition C{k) = (XjICcp' ), k 1, (6.14) one immediately obtains (ij) lb ( (6.15) Following this strategy, the pkrogram efficiently constructs the p + 1 Following this strategy, the program efficiently constructs the p + 1 69 possible reduced Hamiltonian and overlap matrices, from which the secular th solution gives p + 1 roots, of which the i one is selectively reduced. As a bonus the approximate solutions are given, and as our computations will indicate, these are likely to be sufficiently close to constitute a satisfactory treatment for many applications. To place the computational benefits of the reduced partitioning theory into perspective, it should be noted that in any truncated CI cal culation with a basis set, the calculation of the molecular integrals is the predominant time consuming operation, with a considerable amount of time also required for their transformation to a secondary basis. Sine. both of these steps are necessary despite the number of configurations ; used in the calculation, it seems logical to get maximum benefit from the chosen basis by using many more configurations than is customary, or even all that are possible. To do this in the standard approach would involve solving an N x N secular equation for several states, which is, at pres ent, an impossibility for very large N. On the other hand, the reduction process does not require such a diagonalization, but, as will be shown, gives a result in essential agreement with the full solution by only solving low order secular equations. It is still generally necessary to construct all the Hamiltonian matrix elements, but even though this is admittedly a problem for very large N, since these elements are obtainable solely from the transformed molecular integrals and the configurations, the extension is reasonable. VI.2. Results and Discussion: The Single Partitioning Case As a first example of the reduced partitioning procedure (RPP), we considered the h.drogen molecule at the equilibrium internuclear separation *All the calculations reported herein were done on the IBM 360/65 computer of the University of Florida Computer Center. 70 R = 1.40 a.u. The nparticle basis, Ipo h), was generated from an ini tial set of nine STO's on each atom, ls(1.20), Is'(1.00), 2s(1.16), 2po(1.71), 2p +(1.71), 3do(2.20), and 3d (2.00), which, using Schmidt orthogonalized combinations, led to 34 possible configurations of 1e g symmetry. As this is the full CI basis, it gives the best answer obtain able for the chosen space. The full CI result is E = 1.1712787 a.u. The principal contribution to the CI wavefunction is the Coulson ** function configuration To =)(1s o(i) is g(2)) (6.16) which constitutes 98% of the full solution and gives E = 1.1280771 a.u. by itself. Choosing this configuration as the reference function, the {ck) and the ({.i were calculated from (4.44) and (4.45), and from these quantities the reduction energies were obtained from (4.51) for each m from 1 to 13. These numbers are listed in Table 1. For comparison, the difference between the unperturbed energy and the full CI result will be defined as AE(i) = E. (6.17) 1 1 where i indicates the state of interest. If 8. is a HF energy, then AE may be referred to as E the correlation energy obtainable within the corr space if), by analogy with the usual definition of correlation energy. The most dramatic feature illustrated in Table 1 is the anticipated extreme divergence of the sigma series which, lacking a nonlinear summa tion technique, would be useless. However, by applying the [M,M1] Pade *The orbital exponents for n = 2 were taken from the optimized basis set for R = 1.40 a.u. obtained by McLean, et al. [59]. **In fact, Coulson [60]found the best scale factor equal to 1.197 for R = 0.732 A. TABLE 1 Reduction Energies for H2 with the Coulson Reference Functiona (R = 1.40) S (m) = + /( MM] E BrillouinWigner o 8 corr Sigma Elements 1.1280771 1.1582136 1.1662436 1.1696479 1.1707700 1.1711049 1.1712244 1.1712552 1.1712685 1.1712741 1.1712764 1.1712778 1.1712782 1.1712784 33(Full CI) 0.00 69.76 88.35 96.23 98.82 99.60 99.87 99.95 99.98 99.99 99.99 100.00 100.00 100.00 1.1712787 0 = 0.124303 (0 = 0.279955 02 = 1.594896 03 = 1.25 x 101 04 = 1.18 x 102 43 S= 1.22 x 103 5 23 2 = 2.23 x 10 25 aEnergies and bond distances are always expressed in atomic units. approximants to the series, it is seen that the reduction energies con verge very rapidly toward the full CI solution [61]. From the connection between the [M,M1] approximant and the variational treatment, it is more easily seen why this result is found [61]. This is the same situation previously observed in an application of BW perturbation theory to the hydride ion [62], where Pade approximants were also used to sum the series to give meaningful energies. As has been mentioned, the fact that the perturbation functions JcPoc1,...,"c) rapidly exhaust the space of interest causes problems with near linear dependency. In the examples studied, this usually be comes apparent in the ninth or tenth order solution. Consequently, mul tiple precision was used to obtain the higher order solutions in Table 1, permitting another one or two digits of agreement. Due to the simplification inherent in the first order solution, it is especially encouraging that this result accounts for almost seventy per cent of the possible energy improvement. For many problems it may be difficult to calculate the higher order terms, thus a dramatic energy de crease from the "geometric" result is a definite asset. In order to consider a HFR reference function, the previous STO basis set was used in a HFR calculation to give a set of oneelectron orbitals as eigenfunctions to the Fock Hamiltonian. From these, another 34config uration CI wavefunction was constructed but with the principal configura tion now actually being the HF result. The energy of the HF solution is 1.1335224 a.u., and, of course, the full CI energy is the same as before. For a molecule such as H2, where the ls STO combination is a good approx imation to the 1r molecular orbital, there cannot really be too much dif g ference between the Coulson function and the HF solution. This is reflect ed in a comparison of Table 1 and Table 2. The .elements of the sigma TABLE 2 Reduction Energies for H2 with a HartreeFock Reference Function (R = 1.40) M (m) = + l/ (m)[M,M] % Ef BrillouinWigner o + corr Sigma Elements 1.1335224 1.1590861 1.1668000 1.1696330 1.1707910 1.1711357 1.1712490 1.1712697 1.1712763 1.1712779 1.1712785 1.1712786 1.1712786 0.00 67.71 88.14 95.64 98.71 99.62 99.92 99.98 99.99 100.00 100.00 100.00 100.00 33(Full CI) 00 = 0.106462 01 = 0.243929 02 = 1.509162 03 = 1.26 x 101 04 = 1.22 x 102 05 = 1.28 x 10 5 = 1.27 22 023 = 1.27 x 10 23 1.1712787 expansion are very similar as are the reduction energies for each order, although the results with the IF reference function do seem to converge a little better in higher order. The energy decrease given by the geo metric solution is somewhat better for the Coulson function, which is probably a result of more improvement being possible. It should be ob served that better than ninetyfive per cent of the possible correlation energy is given by only a third order or fourfunction expansion. For the HeH molecular ion a very good HF calculation, including orbital exponent optimization, has been done by Peyerimhoff [63]. For R = 1.455 a.u., the optimized basis she obtained is s He(1.37643), s'H (3.87107), 2s (1.54335), 2p (2.64576), 2p' (3.24082), 3d He He o He o He o H (2.54147), 4fo He(3.73526), lsH(1.00949), 2sH(1.18036), 2s'H(2.56229), 2po H(1.79089), 3do H(2.41228), from which, she reports an energy of EHF 2.933126 a.u. By repeating this HF calculation, we obtained the slightly higher result, EHF = 2.933072 a.u., which we expect is due to our more accurate integral computation. By augmenting this sigma basis with 2p He (2.868), 2p l H(0.827), and 3d 2 He(2.686), a full CI wavefunction con sisting of 82 configurations of 1E symmetry was obtained. With the HF solutions as the reference function, the series elements and the reduction energies are as given in Table 3. It appears the sigma series is even somewhat more divergent than in the H2 case which might be expected from the convergence criterion given in (IV.3), but may also be dependent on the adaptibility of the basis set, or the number of configurations. In keeping with the greater extent of divergence, the convergence of the re duction energies is slightly worse than before, even though the third order solution still accounts for more than ninety per cent of the available correlation energy. The geometric result gives about fiftynine per cent. *This STO basis will be referred to as basis I. TABLE 3 Reduction Energies for HeH with a HartreeFock Reference Function (R = 1.455) (m) f BrillouinWigner M ) = 8 + 1/ (m)[M,M1] % Ef BrillouinWigner o corr Sigma Elements io 2.933072 2.956818 2.966665 2.969824 2.971765 2.972594 2.972941 2.973046 2.973105 2.973138 2.973153 2.973162 2.973168 81(Full CI) 0.00 59.21 83.77 91.64 96.48 98.55 99.41 99.68 99.82 99.90 99.94 99.97 99.98 2.973176 T0 = 0.260175 ,1 = 1.723745 2 = 3.01 x 101 3 = 6.58 x 102 330 0 = 1.64 x 10 a = 4.39 x 105 5 0 = 5.60 x 1030 23 4 To obtain a more balanced basis set for potential curve calcula tions for HeH basis II was obtained by excluding the T and 6 functions, as well as the 4f (He) and 2s'(H) functions, which are least important to the HF solution. These were replaced by 2p He(3.00),. 2p~i H(2.00), 3d 2 He(3.00), and 3d2 H(2.00). This basis results in 61 configurations of 1 E symmetry. The HF energy at R = 1.460 a.u. is 2.93302 a.u. In Figure 1, the HF potential curve and the full CI curve for the X E ground state can be compared. Since He is a closed shell atom, this is one of the few special cases when the HF potential separates properly, giving a HF atom and a proton. The main feature to be observed is the relative constancy of the correlation correction. In Table 4 the reduc tion energies through the eighth order are given as a function of R. From the percentages of the correlation energy listed, the convergence with only nine functions is seen to be quite satisfactory, being poorest at R = 2.50 a.u. and excellent at large internuclear separations. The first order solution accounts for better than seventy per cent for larger R values. Figure 2 illustrates this as well as the additional improvement given by the second order solution. It should be noted also that the cor relation energy is not actually constant but shows a rise at R = 1.52 a.u. and decreases slightly as R increases. The question of the shapes of the potential curves for the reduction energies may also be of importance if one is interested in the vibrational spectra. Thus, to more adequately assess the differences between the curves in Figure 2, a Dunham analysis was applied to each of these approx imations using a program written by Beebe [64]. These results are pre sented in Table 5, along with some comparison values obtained in other calculations by Peyerimhoff [63], Anex [65], Michels [66], and Wolniewicz [67]. Since it is not necessarily obvious just how to treat the separated TABLE 4 Reduction Energies for the X + State of HeH as a Function of Internuclear Separationa (Single partitioning with a HF reference function) R 1.00 1.40 1.46 1.52 g(m) (1) (2) (3) (4) 8 (5) g(6) (7) (8) 2.86018 2.88522 2.89392 2.89705 2.89789 2.89807 2.89814 2.89817 2.89818 2.93249 2.95771 2.96703 2.97071 2.97179 2.97205 2.97218 2.97223 2.97226 2.93302 2.95820 2.96757 2.97130 2.97243 2.97270 2.97283 2.97289 2.97292 2.93240 2.95753 2.96693 2.97070 2.97188 2.97217 2.97231 2.97237 2.97240 E (Full CI) 2.89818 2.97228 2.97295 2.97243 o AE 0.03800 0.03979 0.03993 0.04003 %AE(G() 65.89 63.38 63.06 62.78 %AE((8) 100.00 99.95 99.92 99.93 aAll energies are nri.;,aive. TABLE 4 (continued) R 2.00 2.50 3.50 5.00 (m) &0 g(1) (2) e(3) (4) 8 8(5) e(6) &(7) 5(8) Eo(Full CI) 0 2.91057 2.93533 2.94456 2.94835 2.94964 2.94994 2.95011 2.95021 2.95026 2.95033 2.88776 2.91243 2.92112 2.92454 2.92557 2.92578 2.92592 2.92600 2.92604 2.92612 2.86746 2.89242 2.90039 2.90304 2.90371 2.90387 2.90393 2.90395 2.90396 2.90398 2.86218 2.88739 2.89507 2.89752 2.89808 2.89819 2.89822 2.89822 2.89823 2.89823 AE 0.03976 0.03836 0.03652 0.03605 %AE(1() 62.27 64.31 68.35 69.93 %AE(e(8) 99.82 99.79 99.95 100.00 TABLE 4 (continued) R 6.00 8.00 10.00 g(m) e0 5(1) (2) (3) 8 e(4) (5) e(6) (7) (8) Eo(Full CI) 2.86169 2.88694 2.89458 2.89702 2.89756 2.89767 2.89769 2.89769 2.89770 2.89770 2.86147 2.88673 2.89435 2.89680 2.89733 2.89743 2.89746 2.89746 2.89746 2.89746 2.86142 2.88668 2.89430 2.89678 2.89727 2.89738 2.89740 2.89741 2.89741 2.89741 AE 0.03601 0.03599 0.03599 %AE((1)) 70.12 70.19 70.19 %AE(e(8) 100.00 100.00 100.00  4 Ca 4J *HQ 0 4I 0 4J 0 Scd Cd to 0 S rl 0 o 0 00 bO 0) C! r4 aFK1 o ,0 11 %., c\i c \ 10 co U)1 c'Y3 ,I  03 o LUl S a4 q. 0 4J r 0 >. + 4 o CdB 4i ) tiS) 00 4) 0 A n M *i c o 4J 4I 0 l d L41 4J CN 0 W 0 0 44 to ; ' M UO 3 C o o H 0 *to 0 ," o 0 a) S ) wl p * 0 > Q)0 0 o 0 4 P 0 ;0 0 H !i + 8 C0+ C0 I  I I I I 0 I I I I I I  I I I SCO c C\j CN1 Q0 I 0 Q 0 I 0 " " c  LLI 84 atom limit for the &( and (2) curves, the values calculated at 10 a.u. were assumed to be a sufficiently close estimate to use in the deter mination of the dissociation energies. Anex and Wolniewicz used the exact value for the He atom, E = 2.90372 a.u., while in Michels' study, the calculated value of E = 2.87574 a.u. was the reference point. The values for 10 a.u. should be very close to the calculated atom limit, but even though our dissociation energies seem to be somewhat greater than the comparison values, if any error is present, the magnitude should be slightly underestimated. This is not unexpected, however, since bind ing energies defined in this manner are not governed by the variational theorem. The lowest total energy found is that of Wolniewicz where E = 2.97867 at the minimum given in Table 5. Anex's result is E = 2.97424, while Michels reports a value of E = 2.94373 a.u. From the previous spectroscopic parameters in Table 5, it is indi cated that despite the dramatic energy decrease, the actual shapes of the potential curves for the first and second order reduction energies appear to be somewhat further from the "best" results than is the HF curve. This effect of some energy improvement destroying agreement with other proper ties is not uncommon in varieties of perturbation and CI studies. In addi tion to the shape of the potential curves, another manifestation of this lies in the accuracy of the density distribution which is usually apparent from predictions of the dipole moment or other oneelectron properties. Although the HF wavefunction is known to have certain stability fac tors [8], for more general cases these questions have not been adequately pursued. However, one can typically expect better agreement for proper ties more sensitive than teenarwhen additional flexibility is built into the trial wavefunction. Since these low order reduction solutions have +a 44 0 4J 0U 0l (3 uo 0 0 o 41 03 O rt H o C 0 ( 0 0C 0 1 Oi 41 o f 0 CO 0 41 I S I a) u [I 0 ', 0 o* > S3 r> '6 aI C' ,4 0 'A ON 10 N 10 r 0o o 0I 0N C 0 0 10 10 C d) \D Li 14 ,0 r ro rL 0 0 00o co 0 '< 10 Ca C M CM N o o co ai Co CM 10 10 co ai 0 aI ii '.o r '0 4 co L 1o0 10 10 vo Vn V) o 0 10  . 0  1 11 toj T aI aI aI r0" r 04  F Co * *I  '.0  4  C i '%D 1n IN O 1 0o 0 aI aI NM 0 0 o o* r u 0 *I o rc o i4 o A r H4 0 00 0 I o i * I I I 0; i F I N I 0 I * I In I en) C I 10) .0 I . aI I 10 0 0 0 u 141 4.4 4444 w ( 0 0 pag a ex (3 U 1 86 been obtained with only one and two variable parameters, the variational principle apparently overly prejudices the calculation toward an energy reduction with a consequent sacrifice in the accuracy of the potential curves. To avoid these problems, somewhat higher orders must be included. As these are incorporated, the near equality of the eighth order solution and the full CI necessitates that the spectroscopic parameters be essen tially identical. In all of the examples presented in this section,a dramatic energy decrease was given by the low order solutions and essential agreement in the higher orders. In each case, the reduction objective was met with 8 to 12 functions, even though the number of configurations varied from 34 to 82. From the nature of the procedure, it is expected that a similar number of functions should be adequate even with many more configurations. Since we can have only single and double excitations in the systems studied, and the first order solution relative to a IIF reference function depends only on the latter, it may be argued that the energy reduction obtained from the geometric formula may not be as pronounced for larger molecules. However, since it is generally conceded that from an energy criterion the double excitations are the most significant, we still an ticipate a substantial improvement over the HFR result even for other molecules. To obtain reliable spectroscopic properties, though, it is important to obtain slightly higher order solutions. This feature should also apply to observables that are determined by averaging over the electron density. VI.3. Results and Discussion: The Selective Reduction Procedure for Multiple States In order to examine the selective reduction treatment for excited states, the previous 61 configuration, basis II calculation was parti tioned to give a tenfunction reference space composed of the HF solution and the nine possible singly excited configurations. By solving a CI problem with these ten functions, a set of unperturbed solutions, I) , were obtained. From these reference functions, the selective process of (V.2) was used to calculate reduction energies for the ground state and the four lowest excited states of 1 symmetry. The selectivity of the reduction technique is demonstrated in Table 6. Consistent with the steepest descent argument, the percentages of AE(i) indicate that for each choice of unperturbed solution only the cor responding root is substantially affected. The most efficient reduction is found in the ground state, although the reduction process for the low est three excited states is also effective. In each of these cases, the first order solution is still seen to be significant. The ground state perturbation solution obtained with the multi dimensional reference space is interesting, since,by Brillouin's theorem, the single excitations cannot mix with the HFR ground state, audleave one with exactly the same unperturbed energy as in the single partitioning case. Since more functions are explicitly employed in the multidimensional approach, the increased flexibility must of necessity yield some improve ment over the single partitioning case for each order. However, the energy decrease is found to be negligibly limited to the fifth decimal place, thus the results in Tables 4 and 5 are the same for all practical purposes. This consequence attests to the fact that the extent of interaction :nonig the elements of the reference space and the contribution from the TABLE 6 Illustration of Selective Reduction for Multiple States of HeHa (Tenfunction partitioning; HF and single excitations; R = 1.460) Unperturbed o Energy Reference Function o (1)) %AE(i) 8(8) %AE (i) (1) %AE(i) (8) %E (i) 8(1) %AE(i) 8(8) %AE (i) 8(1) %AE(i) 8(8) %AE(i) 2.93302 1.91869 1.41766 1.10141 0.61147 2.95830 63.31 2.97294 99.97 2.93445 03.58 2.95028 43.22 2.93651 08.74 2.93887 14.65 2.93630 08.21 2.95176 46.93 1.91938 00.83 1.96910 60.95 1.95689 46.19 2.00131 99.89 1.91977 01.31 1.92574 08.52 1.92222 04.24 1.93000 13.67 1.41900 02.13 1.41954 02.99 1.41845 01.26 1.42300 08.50 1.44910 50.06 1.48031 99.76 1.41778 00.19 1.43357 25.33 1.10398 04.79 1.12804 49.62 1.10753 11.40 1.11040 16.75 1.10166 00.47 1.11245 20.57 1.12206 38.48 1.15461 99.12 0.61191 00.27 0.69951 53.52 0.61159 00.07 0.73366 74.28 0.61164 00.10 0.62182 06.29 0.61147 00.00 0.62356 07.35 aAll energies are negative. TABLE 6 (continued) i 0 1 2 3 4 0 (1) 2.93392 1.91880 1.41779 1.10141 0.64363 4 %7E(i) 02.25 00.13 00.21 00.00 19.55 (8) 2.94321 1.96909 1.42581 1.11437 0.73923 %AE(i) 25.52 60.94 12.98 24.15 77.67 Full CI (E.) 2.97295 2.00140 1.48046 1.15508 0.77597 Solution i orthogonal complement is insignificant for this state compared to the mixing of the ground state unperturbed solution, i.e., the HFR result, and the perturbation functions derived from it. Since basis II is very prejuduced toward the ground state, of the four excited states, only the first, the A + state, behaves correctly at large internuclear separation, where it converges reasonably well to a + 1+ He (1s) + H(ls) separated atom limit. The relationship between the X I and A curves is illustrated in Figure 3. Although it is not apparent in the figure, from the full CI results listed in Table 7, the first excited state actually shows a slight minimum in the vicinity of R = 6.5 a.u. This feature has been previously observed by Michels [66] who more cor rectly positions the minimum at 5.65 a.u. In Figure 4 the first and second order reduction energies are com pared with the unperturbed result and the full CI solution. As before, the extent of reduction given by the low order solutions is very satis factory. In fact, (2)(R) apparently converges to the same separated atom limit as the full CI, although (1)(R) does not seem to be able to do so. The eighth order result is observed to be essentially identical to the full solution. As has been mentioned, the approximate quantities defined in (5.32) are obtained as an intermediate in the computational procedure. In order for these conceptually appealing results to yield reasonable estimates, the interactions arising from the remaining p unperturbed solutions need to be small as was indicated from the ground state discussion. If so, neglecting the mixing should not appreciably affect the spectrum of the results. That this is actually the case in this example is illustrated in *A correlation diagram for the lower states of Hell has been given by Michels [66]. 4 0 .4" 00 0 00 C 0 C4 ) .0  to 0 t o01 * OO4 N N *t N CM CM 0 o 4J cc 0 01 a) 1 o .c, 44 0 0) 4 4 0 Om r4 d c 0 4 1 0 40 0 o r4 41 U U 0X o o 0u O) bo 44 *J 4 H 41 Q) S0 0 .1 S4 ci) ffi) 0 0 Oc O 0 OC Lr) 0 * 0 Co O o * 00 cc N oC CO C 01 CO a n 11 r4 '4 r< 3 r1 (d CM( O 3 sAe cc 0 ,c r s d .4 ] wi 4 (*O cO BS &l B^ c. s^ M N 0 \0 O O 0 4 CM 00 c0 0M CM CN N 