Upper and Lower Bounds for GroundState SecondOrder Perturbation EnergyStephen Prager and Joseph O. Hirschfelder Citation: The Journal of Chemical Physics 39, 3289 (1963); doi: 10.1063/1.1734192 View online: http://dx.doi.org/10.1063/1.1734192 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/39/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A rigorous lower bound for the groundstate energy of the hydrogen atom Am. J. Phys. 58, 554 (1990); 10.1119/1.16448 Lower Bounds to the GroundState Energy of Systems Containing Identical Particles J. Math. Phys. 10, 562 (1969); 10.1063/1.1664877 Improved Equation for Lower Bounds to Eigenvalues; Bounds for the SecondOrder Perturbation Energy J. Chem. Phys. 50, 2758 (1969); 10.1063/1.1671442 Variational Upper Bounds to SecondOrder Energies for Excited States J. Chem. Phys. 49, 1411 (1968); 10.1063/1.1670241 Upper and Lower Bounds on the SecondOrder Energy Sum J. Chem. Phys. 47, 2707 (1967); 10.1063/1.1712287

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 39, NUMBER 12 15 DECEMBER 1963

Upper and Lower Bounds for Ground-State Second-Order Perturbation Energy*

STEPHEN PRAGER

Department of Chemistry, University of Minnesota, Minneapolis, Minnesota

AND

JOSEPH O. HIRSCHFELDER

Univers1ty of Wisconsin, Theoretical Chemistry Institute, Madison, Wisconsin

(Received 9 August 1963)

The first-order perturbation equation, using the Dalgarno-Lewis formulation, is arranged in a form analogous to the Poisson equation for the electrostatic potential produced by a charge distribution in a medium of variable dielectric constant Thus, the Thomson and Dirichlet variational principles of electro­statics can be used to obtain approximate solutions to the first-order perturbation equation for systems in either the ground state or the lowest energy state of a given symmetry. The Thomson principle provides a useful lower bound to the second-order perturbation energy. The Dirichlet principle is derivable from the Rayleigh-Ritz or Hylleraas principles and gives an upper bound to the second-order energy. For excited states, the Sinanoglu principle provides the upper bound. By optlmizing the scaling of the trial perturbed wavefunction, the Hylleraas principle is presented in a somewhat improved form. As an example, the polarizability of atomic hydrogen is used to illustrate both the Thomson and Dirichlet principles and to place upper and lower bounds on the polarizability

I. INTRODUCTION

THE first-order perturbation equation in molecular quantum mechanics can be expressed in a form

which is similar to the equation for the electrostatic potential of a charge distribution in a medium of variable dielectric constant. The second-order perturba­tion energy is proportional to the electrostatic energy of this charge distribution. By this analogy, the well­known Dirichlet and Thomson variational principles of electrostaticsI,2 can be used to provide upper and lower bounds for the second-order perturbation energy of a molecule in its ground state. The Dirichlet principle for the upper bound is derivable from the quantum mechanical Rayleigh-Ritz principle and is somewhat better than the Hylleraas principle3,4 which is fre­quently used for perturbation problems. The Thomson principle is our only means of obtaining a lower bound to the second-order perturbation energy without explicit knowledge of the energy of excited states. The Weinstein-MacDonald principle4.5 provides a lower bound for the energy of the system but it does not even

* This research was supported by the following grant: National Aeronautics and Space Administration Grant NsG-275-62 (4180).

1 R. Courant and D. Hilbert, Methods of Mathematical Physics, (Interscience Publishers, Inc., New York, 1953); 1st English ed., Dirichlet principle: Vol. I, p. 240; Thomson principle: Vol. I, p.267.

2 D. M. Schrader and S Prager, J. Chern. Phys. 37, 1456 (1962) .

3 E A. Hylleraas, Z. Phys. 65, 209 (1930). 'E. A. Hylleraas, "The Variational PrinCIple in Quantum

Mechanics," Rept. No.1, Institute for Theoretical Physics, University of Oslo, Oslo, Norway (1961).

6 D. M Weinstein, Phys. Rev. 40, 797 (1932) and 41, 839 (1932); Proc. Natl. Acad. Sci. (U S) 20, 529 (1934). J. K L. MacDonald, Phys. Rev. 46, 828 (1934).

give a correct first-order perturbation energy, and therefore it is useless in predicting the second-order energy. The Thomson principle seems to be comparable to the Temple4 and better than the Dalgarn06 prin­ciples even though the Temple and the Dalgarno rela­tions make use of the energy of the first excited state (perturbation coupled to the ground state). The calcu­lation of the polarizability of atomic hydrogen is used as a simple example to show the efficacy of these principles.

Following Sinanoglu,7 we can extend the Dirichlet principle to give upper bounds for the energy of mole­cules in excited states by making use of the unperturbed wavefunctions for each of the lower energy states. The Thomson principle might be extended to give lower bounds to the energy for excited states when the perturbation equation is separable. However, when the perturbation equation is separable, the solutions may be obtained by quadrature and there is little need for a variational principle.

The Dalgarno-Lewis procedure8,9 for determining explicit solutions to the perturbation equations has made it feasible to apply perturbation theory to a wide class of molecular problems where complete sets of solutions are not known for the unperturbed sys-

6 A. Dalgarno, Quantum Theory, edited by D. R. Bates (Aca­demic Press Inc., New York, 1961), Vol. I, Chap. 5, Secs. 2.19 and 2.20 on p. 191.

7 O. Sinanoglu, Phys. Rev. 122, 491 (1961). 8 General perturbation theory A. Dalgarno, Quantum Theory,

edited by D. R Bates (Academic Press Inc., New York, 1961), Vol. I, Chap 5; C Schwartz, Ann. Phys. (N.Y) 2, 156, 170, and 176 (1959); A Dalgarno and J T. Lewis, Proc. Roy. Soc. (Lon­don) A233, 70 (1956); A. Dalgarno and A L. Stewart, ibid A238, 269 (1956).

9 J. O. Hirschfelder, J. Chern. Phys. 39, 2099 (1963).

3289

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3290 S. PRAGER AND J. O. HIRSCHFELDER

tem.10 Perturbation treatments have the advantage over variational methods that they permit the system itself to select the proper type of terms which should be included in the trial wavefunction. From a knowl­edge of the trial wavefunction through the first order, the energy can be calculated accurately through the third order. For many chemical purposes this is suffi­cient accuracy.

Dalgarno and Lewis8 •9 express the first-order per­turbed wavefunction as a product of a function F and 1/10. Here 1/10 is the zeroth-order (or unperturbed) wave­functionll for the state under consideration (which is designated by the subscript "0"). For convenience we take 1/10 to be real. If V is the perturbation potential, the first-order perturbation equation can be expressed in the form

Here fo(l) is the first-order perturbation energy,

(2)

The second- and third-order energies, fO(2) and fo(3), are

The product F1/Io must satisfy the same boundary con­ditions and the same conditions of continuity, square­integrability, and continuity of the first derivatives (except at interior points where the potential energy is singular) as are normally expected of stationary­state wavefunctions. Equation (1) defines F except for an arbitrary additive constant. Usually this constant is adjusted so as to make NoF1/Iodr=0. If 1/1. is the un­perturbed wavefunction for the state "s" and the un­perturbed energies are f. and fO for the two states,

ture if the state under consideration is either the ground state or the lowest energy state of a given symmetry. For other states, difficulties are encountered since F may have poles at points where 1/10 has nodes. For ex­cited states, there are special procedures for deter­mining F provided that Eq. (1) is separable.12 Except in Sec. VI, we confine our attention to the ground state or the lowest energy state of a given symmetry. The higher-order perturbation equations can be written in the same general form as Eq. (1). Therefore, without difficulty, the results of the following sections can be generalized to the higher orders of perturbation.

The first-order perturbation relation, Eq. (1), may be regarded as a Poisson equation

V· (KVcf» = -471"p, (6)

for the electrostatic potential cf>= F produced by the charge distribution p= - (271")-11/10(V -fO(1))1/10 in a me­dium of variable dielectric constant K = 1/102• Further­more, the electrostatic self-energy of the charge dis­tribution is U =! f pcf>dr= - (471") -lfo(2). Thus, the Thom­son and Dirichlet principles of electrostatics can be applied to obtain lower and upper bounds for fO(2) pro­vided that singularities in F arising from nodes in 1/10 are avoided by restricting the discussion to the ground state or to the lowest energy state of a given symmetry.

For simplicity of presentation and to make the elec­trostatic analogy closer, the quantum mechanical rela­tions are presented for a one-electron system. However, all of the quantum mechanical relations can be gen­eralized to a system of N electrons by considering all vectors to be 3N dimensional and the symbol V to be the gradient in the 3N-dimensional space of the N electrons.

II. THOMSON AND DIRICHLET PRINCIPLES IN ELECTROSTATICS1,I

In a medium of variable dielectric constant K, the self-energy U of a charge distribution p may be written in a number of different ways13:

(5) U=! !pcf>dr= (871")-1/ KVcf>,Vcf>dr= (871")-1/ KE·Edr.

If Eq. (1) is separable, F can be determined by quadra-

10 Atomic polarizabilities and shielding factors: A. Dalgarno, Advan. Phys. 11, 281 (1962). Magnetic susceptibility of diatomic molecules: M. Karplus and H. J. Kolker, J. Chern. Phys. 38, 1263 (1963). Van der Waals forces: A. Dalgarno and A. E. Kings­ton, Proc. Phys. Soc. (London) 78,607 (1961) and 73, 455 (1959) ; also A. Dalgarno and N. Lynn, Proc. Roy. Soc. (London) A69, 821 (1956) and A70, 223 (1957). First-order solutions of atomic Hartree-Fock equations: M. Cohen, Preprint No. 42, Quantum Theory Project, University of Florida, April 1963.

11 The stipulation that ,yo is the zeroth-order wavefunction im­plies that the state under consideration is either nondegenerate, or if it is degenerate, the degeneracy is removed by the first-order perturbation. Other degenerate problems require special treat­ment (see Refs. 8 and 9). If the states 0 and 0' have the same unperturbed energy, then I,yo', F,yodr=O.

(7)

Here the electrical field E is minus the gradient of the

12 Solution to perturbation equations for excited states. W. Byers Brown and J. O. Hirschfelder, Proc. Nat!. Acad. Sci. (U.S.) 50,399 (1963).

13 If K is constant,

<p(rI) = K-IIrI2-Ip (r2)dr 2 and

U = (2K) -11 frt2 -lp (rI) p (r2) drldr2.

If, however, K is not constant, we do not know how to determine <p(rl) in terms of a quadrature. Instead, we can give an integral equation which might prove useful,

<p (rI) = IrI2-IK (r2)-I[p (r3) + (h)-IV3K (r2) • V2<P (r2) Jdr 2.

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B 0 U N D S FOR G R 0 U N D - S TAT E PER T U R BAT ION ENE R G Y 3291

electrostatic potential,

E=-vrp. (8)

If, in place of the true potential rp [which satisfies Eq. (6) ] and the true electric field E [which satisfies (8) ] we use the trial potential cp and the trial field E tr, the three integrals in Eq. (7) are no longer equal nor equal to U. However, we have the following inequalities.

(1) Thomson's principle states that the self-energy given in terms of the trial electric field constitutes an upper bound on U,

(9)

subject to the condition that the trial electric field satisfies the Poisson equation in the form

(10)

The equality in (9) only applies if E tr also satisfies Eq (8), in which case the trial field is equal to the true field.

(2) Dirichlet's principle provides a lower bound to the self-energy. It states that

U~21r[fpCPdrT / f KVcp·Vcp·r (11)

for any trial potential cp that goes to zero at infinity. The equality in (11) is only obtained when cp satisfies Eq. (6) and is therefore equal to the true potential.

Equations of the Poisson type occur in connection with a wide variety of physical phenomena. Thus, the Thomson and Dirichlet principles have been used to determine the magnetic permeability in multiphase systems,14 the effective diffusion constant in poly­electrolyte solutions,15 the Fermi-Thomas energy of atomic systems,16 and the evaluation of quantum mechanical exchange and coulombic integrals.2 In addi­tion, the Dirichlet principle has been used to deter­mine the Brownian movement in many-particle sys­temsI7 and the diffusion and viscous flow in concentrated suspensions.Is With such a history of successful applica­tions, it is not surprising that the Thomson and Dirich­let principles can be usefully applied to perturbation problems.

III. LOWER BOUND FOR EO(!): THOMSON'S PRINCIPLE

Thomson's principle can be applied to perturbation theory provided that neither 1/10 ( V - EO (1) 1/10 nor F have singularities. The first condition is satisfied if the Rayleigh-Schrodinger perturbation theory is applicable

1( Z. Hashin and S. Shtrikrnan, J. App!. Phys. 33, 3125 (1962). 16 J. L. Jackson and S. R. Coriell, J. Chern. Phys. 38, 959

(1963) . 16 O. B. Firsov, Zh. Eksperirn. i Teor. Fiz. 32, 1464 (1957)

[Eng!. Transl: Soviet Phys.-JETP 5, 1192 (1957) J. 17 G. Woodbury and S. Prager, J. Chern. Phys. 38,1446 (1963). 18 S. Prager, Physica 29, 129 (1963); see also AFOSR Tech.

Rept. No. 2803 (June 1962).

(without special considerations). The second condition is satisfied if the state "0" under consideration is the lowest energy state of a given symmetry. Under these conditions, by analogy with Eqs. (9) and (10),

(12)

subject to the condition that the trial "field" Gtr

satisfies the Poisson equation in the form

V· (-.f;02Gtr) =-21/10(V-Eo(I»1/Io. (13)

The equality in (12) applies if, and only if, G tr= - V F. Proof: First, let us prove that the equality in (12)

applies for the true "field" G= - V F. Multiplying Eq. (1) by F and integrating over all space,

! f Fv· (1/I02VF)dr= f%(V-eo(I»1/IoFdr=EO(2). (14)

The conditions for the applicability of Thomson's principle, stated above, ensure that Gauss' theorem can be applied to the left-hand side of Eq. (14),

! f FV· (%2VF)dr=! !%2FVF.dS-! !%2VF.VFdr.

all space boundary surface at infinity

all space

(15)

The surface integral in Eq. (15) vanishes because of the boundary conditions imposed upon the stationary­state wavefunction % and its perturbation F% (they go to zero faster than any negative power of the distance as the distance approaches infinity). Combining Eqs. (14) and (15) and using the definition of G,

(16)

Furthermore, Eq. (1) can be written in terms of G,

V· (%2G) = - 2%(V -EO(I»%. (17)

Now let us consider a trial field Gtr which differs from the true field,

(18)

Here oG may be arbitrarily large. Substituting Eq. (18) into Eq. (16),

EO(2) = -! f%2Gtr'Gtrdr-f%2Gtr,oGdr

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3292 S. PRAGER AND J. O. HIRSCHFELDER

Substituting Eq. (18) into Eq. (17) and imposing the condition (13),

(20)

Multiply Eq. (20) by F and integrate over all space, to give

For completeness, we should also consider the Temple4 and Dalgarn06 variational principles which require knowledge of El, the energy of the first excited state which is perturbation coupled to the ground state. The Temple principle states that

Eo'2. J l - (J2- J12) (El - Jl)-r, (27)

f Fvo (l/Io2oG)dr=0. (21) with the proviso that

If oG has no singularities, Gauss' theorem may be applied to Eq. (21) to give

fl/lo2oGoVFdr=0. (22)

Making use of Eq. (22) and the definition G= - V F,

f l/Io2Go oGdr= f l/Io2[G+ V F} oGdr= O. (23)

Thus, Eq. (19) becomes

EO(2) = -! f l/Io2Gtr oGtrdr+! fUoGo oGdr. (24)

As applied to a perturbation problem,

Eo '2. EO+XEO(I)+X2A +X3( 000) + 00', (28)

where

Thus, the Temple principle gives the lower bound for the second-order perturbation energy of the ground

Since the second integral is necessarily positive, the state, Thomson principle, Eq. (12), is necessarily true.

(30)

Comparison of Thomson with Other Lower-Bound Energy Principles

The best previous principle for obtaining lower bounds for the energy of the ground state (without requiring knowledge of the energies of excited states) is that of Weinstein and MacDonald.4.5 If H is the Hamiltonian for the system, Eo is the energy of the ground state, 1{; is a normalized trial wavefunction, J l = f1{;Hlf/dr, and J2= f1{;H21{;dr, then the Weinstein­MacDonald principle states that

Eo'2.Jl-(J2-N)~. (25)

Let us apply this principle to a perturbation problem where H = ho+X V, the energy of the ground state is Eo=EO+XEo(I)+X2EO(2)+ .. 0, and its true wavefunction is l/Io+XFl/Io+ 000. If we suppose that the unnormalized trial wavefunction is l/Io+ XFl/Io, then the Weinstein­MacDonald principle states that

Eo '2. EO+XEO(I)-x[f eCho-Eo) Fl/Io+ (V -Eo(I))¢'o]2dr T +X2( ... )+ .... (26)

Since the integrand in Eq. (26) is only zero if F is equal to the true function F, the coefficient of the first power of the perturbation parameter on the right-hand side is not XEo(I). Thus, the Weinstein-MacDonald principle does not even accurately predict the first-order energy and gives us no information regarding the second-order perturbation energy in this application.

The Temple principle (30) can be optimized with respect to the magnitude of F to give the improved result

(30')

where

x f F¢'o(ho- Eo)2Fl/Iodr.

By taking F= 0 in (29), we obtain Dalgarno's6 relation for the lower bounds,

(31)

The Temple seems to be considerably better than the Dalgarno principle. However, the Dalgarno relation does not require the use of an approximation to the first-order perturbed wavefunction. In Sec. V, for the

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D 0 U N D S FOR G R 0 U N D - S TAT E PER T U R BAT ION ENE R G Y 3293

example of a hydrogen atom placed in an electric field, we find that the Thomson principle gives a slightly better result than the Temple and a much better one than the Dalgarno relations.

IV. UPPER BOUND FOR '0(2): DIRICHLET'S PRINCIPLE (DERIVED FROM RAYLEIGH-RITZ PRINCIPLE)

Dirichlet's principle can be applied to perturbation theory with the same provisos as given for the Thomson principle in Sec. III. By analogy with Eq. (11),

€o(2)~-2[f~o(V-€o(1))~oFdrT / jNVF'VFdr, (32)

where F is any trial function subject to the conditions that it not have any singularities and that ~o2FV F ap­proaches zero faster than the reciprocal of the square of the distance in the limit as the distance approaches infinity. The equality holds if, and only if, F is propor­tional to the true function F. If the trial function is taken to be aF where a is a proportionality constant, the right-hand side of (32) is independent of a. For any other property depending on the magnitude of F, the most probable value of the constant a is

a=-2f~o(V-€o(1)~oFdr / f~o2vF.vFdr. (33)

Dirichlet's principle is only applicable to molecules in their ground state or the lowest energy state of a given symmetry. Sinanoglu's principle, discussed in Sec. VI, provides an upper bound for the second-order perturba­tion energy of excited-state molecules.

Proof: Although the Dirichlet principle can be justi­fied directly by using standard methods of variational calculus, we derive it in two steps starting with the Rayleigh-Ritz minimum-energy principle. Using the notation of the previous section, the Rayleigh-Ritz principle states that Eo~ J 1. In a perturbation problem with H = ho+X V, if we evaluate J 1 using the unnor­malized trial wavefunction ~o+AF~o, then the Ray­leigh-Ritz principle gives

Eo~€o+X€0(1)+X2Q+A3( ... ) +"', (34) where

Since Eo= €O+A€O(l)+X2€O(2)+ •• " we can subtract €o+A€o(1) from both sides of (34) and divide by A2 to obtain

€O(2) +X€O(3) + ••• ~Q+A("') +.... (36)

Since Eq. (35) is true for all sufficiently small values of A, it follows that

(37)

This is the Hylleraas principle.4 ,5 The equality holds if,

and only if, the trial function F is equal to the true function F. The Hylleraas principle has been used extensively in perturbation theory to provide a per­turbation-variational upper bound on the second-order perturbation energy. The Dirichlet principle, Eq. (32), is readily derived from the Hylleraas principle. How­ever, the Dirichlet principle usually gives a closer limit.

The first step in deriving the Dirichlet principle is to examine the first integral in Q. Since ho= -!V2+vo where vo is the potential energy of the unperturbed system,

And, of course, ho~= €o~o. Thus,

=+! f ~o2VF·VFdr. (39)

The last step in Eq. (39) is obtained through integra­tion by parts and is only permissible if F has no singu­larities, and ~o2FV F approaches zero faster than the reciprocal of the square of the distance in the limit as the distance approaches infinity. Thus, the Hylleraas principle can be restated in the form

Now, in place of F, let us use as trial function aF. Then (40) becomes

€O(2)~~ f ~o2vF. vFdr+2a f ~o(V -€o(l))~oFdr. (41)

By varying the constant "a," the right-hand side of (41) can be minimized and the upper bound for €O(2) improved. The optimum value of "a" is given by Eq. (33). Inserting Eq. (33) into (41) gives the Dirichlet principle as expressed in (32).

V. ILLUSTRATION: POLARIZABILITY OF Is ATOMIC HYDROGEN

The polarizability of a hydrogen atom in its ground state provides a good example to illustrate the applica­tion of the Thomson and Dirichlet principles. Consider the atom perturbed by a uniform electric field in the x direction. The electric field strength is taken to be the perturbation parameter and V = -x. The first-order perturbation energy (or Stark effect) is zero for this case. The polarizability is defined to be a= -2eo(2). The 1s wavefunction for the unperturbed hydrogen atom is ~o= (7r)-t exp( - r).

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3294 S. PRAGER AND J. O. HIRSCHFELDER

Thomson's Principle

Condition (13) becomes for this example,

v· [Gtr exp( - 2r) J= 2x exp( - 2r).

A possible trial field which satisfies Eq. (42) 1S

Gtr= - (!+r)j".

(42)

(43)

Here jx is the unit vector in the x direction. That the trial G tr of Eq. (43) is not the correct field is evident from the fact that the curl of Eq. (43) does not vanish. Thus, Gtr is not equal to -VF. Substituting (43) into (12) gives

or a<4.75. (44)

Dirichlet's Principle

F=ax provides a reasonable trial function for use in Dirichlet's principle, (32). The upper bound for Eo (2) is then

EO(2) < -2[;/ x2 exp( -2r)dr J = -2, (45)

or, a>4. According to Eq. (33),

a= 2(11')-1/ x2 exp( - 2r)dr= 2. (46)

Thus, 4.00<a<4.75. The correct value of a is 4.50. By using more complicated functions for G and F, the limits on a can be made as close as desired.

Temple's Principle

The first perturbation coupled excited state is the 2px with the energy E1=-1/8. Again taking F=ax as a reasonable trial function for use in Temple's principle (30), a lower bound for EO(2) is

The optimum value of a is a= 6/7 so that

EO(2) > -50/21= -2.381 or a<4.762. (48)

The large difference between a= 6/7 for the Temple and a= 2 for the Dirichlet principles should be noted.

Dalgamo's Principle

Dalgarno's principle (31) gives the comparatively poor lower bound

EO(2) > -8/3= -2.667 or a>5.333. (49)

VI. UPPER BOUND FOR EXCITED STATE fO(2):

SINANOGLU'S PRINCIPLE

For excited molecular states, the upper bound for Eo(2) may be obtained from Sinanoglu's principle7 :

EO(2)::; f F1{Io(ho- Eo) F1{Iodr+2/ 1{Io( V -EO(!) F1{Iodr, (50)

subject to the condition that

(51)

for all states "s" which are lower in energy than the "0" state under consideration, EO> E •• The trial function F is further restricted by the requirement that F1{Io satisfy the same boundary, continuity, and integrability conditions required of 1{Io. The equality in (SO) holds if, and only if, F is equal to the true F. The condition expressed in Eq. (51) is just the requirement that the trial wavefunction for the excited state be orthogonal to the states of lower energy (through the first order in the perturbation).

Proof: Proof of the Sinanoglu principle is very similar to the proof of the Dirichlet principle given in Sec. IV. We use for the unnormalized trial wavefunc­tion 1{Io+"A F1{Io. Because of the conditions given by Eq. (51), this trial wavefunction is orthogonal to all of the lower energy states of the system. Because of this orthogonality,4 Eo::; J 1• Thus, with the conditions of Eq. (51), the Hylleraas principle, Eq. (37) or (SO), still applies

For excited states, the Hylleraas principle cannot be rearranged into the form of Eq. (40) since F may have poles at places where 1{Io has nodes and therefore Gauss' theorem may not be used in Eq. (39). Because of the conditions of Eq. (51), a proportionality constant cannot be introduced into the trial function and optimized.

ACKNOWLEDGMENTS

The authors wish to thank the Journal of Chemical Physics referee for many helpful comments. They also wish to thank Kenneth Sando and Daniel Konowalow for checking the validity of the relations for many­electron systems.

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