abdel-raouf. variational methods for bound-state and scattering problems

Abdel-Raouf, On the variational methods for bound-state and scattering problems 165 Abbreviations AFM = Anomaly-Free Method BGVM = Bubnov-Galerkin’s Variational Method BSP = Bound-State

ON THE VARIATIONAL METHODS FOR BOUND-STATE AND SCATTERING

PROBLEMS

Mohamed Assad ABDEL-RAOUF

Fakultat fiir Physik der Universitat Freiburg, D- 7800 Freiburg, West Germany

PHYSICS REPORTS (Review Section of Physics Letters) 84, No 3 (1982) 163-261 North-Holland Publishing Company

ON THE VARIATIONAL METHODS FOR BOUND-STATE AND SCATTERING PROBLEMS

Mohamed Assad ABDEL-RAOUF

Fakultat fiir Physik der Universitat Freiburg, D-7800 Freiburg, West Germany

Received August 1981

Put forward what is true

So write that it may be clear, Fight for it to the end

L Boltzmann

Contents:

Abbreviations 165 3.2.1 Variational methods of first order operators 205

2 Generalized variational and projection formalism 168 3.3 Bounds for the energy and expectation values 217

3 Variational methods for bound-state problems 171 3.3.1 Energy-lower bounds 217 3.1 Variational methods for eigenvalue problems 172 3.3.2 Bounds for the expectation values 245

3.1.2 Quadratic Variational Methods (QVM) 197 Acknowledgment 254

Copies of this issue may be obtained at the price given below All orders should be sent directly to the Publisher Orders must_be |

PHYSICS REPORTS (Review Section of Physics Letters) 84, No 3 (1982) 163-261 North-Holland Publishing Company

ON THE VARIATIONAL METHODS FOR BOUND-STATE AND SCATTERING PROBLEMS

Mohamed Assad ABDEL-RAOUF

Fakultat fiir Physik der Universitat Freiburg, D-7800 Freiburg, West Germany

Received August 1981

Put forward what is true

So write that it may be clear, Fight for it to the end

L Boltzmann

Contents:

Abbreviations 165 3.2.1 Variational methods of first order operators 205

2 Generalized variational and projection formalism 168 3.3 Bounds for the energy and expectation values 217

3 Variational methods for bound-state problems 171 3.3.1 Energy-lower bounds 217 3.1 Variational methods for eigenvalue problems 172 3.3.2 Bounds for the expectation values 245

3.1.2 Quadratic Variational Methods (QVM) 197 Acknowledgment 254

Copies of this issue may be obtained at the price given below All orders should be sent directly to the Publisher Orders must_be |

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 165

Abbreviations

AFM = Anomaly-Free Method BGVM = Bubnov-Galerkin’s Variational Method BSP = Bound-State Problems

CVM_ = Classical Variational Methods HMVM = Harris—Michels Variational Method HVM_ = Hulthén’s Variational Method IAFM = Interpolated Anomaly-Free Method

KI = Kato’s Identity

KIE == Kato’s Inequality

KVM_ = Kohn’s Variational Method LNVM = Least Norm Variational Method LSM = Least-Squares Method

LVM = Ladanyi’s Variational Method MBDM = Minimum-Basis-Dependence Method MEVM = Minimum-Error Variational Method

MIO = Method(s) of Intermediate Operators MNM_ = Malik’s Variational Method(s) NVM =Nesbet’s Variational Method(s) OAFM = Optimized Anomaly-Free Method

OMNM = Optimized Minimum Norm Method

QVM = Quadratic Variational Method(s)

PT = Partitioning Technique QHM = Quadratic Hulthén Method QLSM = Quadratic Least-Squares Method

QKM_ = Quadratic Kohn Method

QRM_= = Quadratic Rubinow Method

= Restricted Int lated Anomaly-Free Method RRM _ = Rayleigh—Ritz Method

i i ? ation, leads to a very complicated situation

from the moment one wishes to bàn vàn problems more complicated than the motion of one electron

approximation, by Schrödinger (1-4) i in his early work “Quantisierung als Eigenwertproblem” and s since

166 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

that time, many authors have used and developed variational methods (VM) for bound-state and

scattering problems (BSP and SP, respectively) Most of these methods have been separately analysed and reviewed with their applications in many books and review articles (see e.g [5-9] for BSP and [10-21] for SP) However, according to our knowledge, a comparative study of both types of methods in the framework of one variational theory has not yet been presented From the historical point of view,

variational methods were utilised from the beginning of the quantum theory, as mentioned above, by

Schrodinger (see eqs (23) and (24) in [1]) only for BSP Twenty years later, Hulthén [22] established the first successful variational techniques for SP It is also important to mention here that, whilst BSP are

mathematically equivalent to the very old eigenvalue problem, a fact which has allowed for much valuable communication between the literature of mathematical and theoretical physics, SP are of more

complicated nature and so far have been treated mostly by physicists

The Rayleigh—Ritz method (RRM; [23-27]) is the first VM of bound-state type It has been applied in the treatment of ground-states of atomic, molecular and nuclear systems and extended, through the

theory of Hylleraas-Undheim [28], to excited states of any quantum mechanical system Trial in-

vestigations of a generalized RRM, for eigenvalue problems composed of two Hermitian observables

(see section 3), are given by Biedenharn and Blatt [29], Michlin [6] and Michlin and Smolitskiy [7]

Other variational methods (quadratic in the Hamiltonian and henceforth we call them quadratic variational methods; QVM) have been suggested by James and collaborators [30,31], by Frost [32], Frost et al [33,34] and others for estimating the eigenvalues and eigenvectors of BSP On the other

hand, the interest in calculating other expectation values of certain nondominating observables (say the

position vectors of some electrons of an n-electron system) appearing in BSP, has led to the

development of variational techniques specifically for these problems However, these methods are

related to RRM (see e.g Delves [35, 36]) and to QVM (see Lloyed and Delves [37, 38]) Further

improvement has emphasized the formulation of variational and supervariational methods (see the

works of Gerjuoy et al [39-43] and Spruch and Rosenberg [44]) which are based essentially on the use

of Lagrangian multipliers

For scattering problems, Hulthén ([22, 45, 46]; HVM), Kohn ((47]; KVM, see also Huang [48—-50])

Hulthén [51] and Feshbach and Rubinow ([52~54]; RVM) have proposed, on the one hand, the so called

classical variational methods (CVM) On the other hand, Schwinger ({55, 56]; SVM) has introduced a

VM based on a completely different idea Immediately following the establishment of the CVM for SP,

nors (e.g h | and Nesbe 0|) were able to show that these method

always associated with the appearance of spurious singularities L7] (anomalies [59] or

to the characteristics of the Hamiltonian of the considered system (see section 4.1.1.6) Malik (6i 62]

and C Michele y O04, x4 4 £ OD]; NM+L VN, Nesbe \ "A 0; Q D „ ĐỒ; O31; a Hy 7 adan Lad y

al 1 (69 70): LVM see also (71-771) and Takatsuka and Fueno ([7S], TFVM) tried, sometimes with less SUCCESS , to construct variational methods which are fr malies T

methods and the CVM as Hulthén-like variational methods.) Absolutely anomaly-free methods, similar

to Hulthén-like methods but with matrix elements involving quadratic Hamiltonians, 1.e QVM, were

presented very recently by Abdel-Raouf [79], for solving SP Nevertheless, much effort has been

invested in extending VM from one of the above mentioned fields into the other For example,

Providica [80] and others (see e.g [81, 82]) have used Schwinger’s version in BSP, Singh and Stauffer

[83] have constructed two variational schemes similar to KVM and SVM for dealing with BSP and stressed their advantages compared to RRM (see also Kolker [84]) Additionally, Obu [85] has

investigated bound-state problems in which the interaction is non-Hermitian (i.e in the region where

RRM becomes inapplicable) He indicated that in these cases one can use either HVM or KVM in their

M.A, Abdel-Raouf, On the variational methods for bound-state and scattering problems 167

original forms if the eigenvalues are assumed to play the role of the phase shift in simple elastic

scattering problems Non-Hermitian operators appear especially in some Bethe-Salpeter problems

[86, 87] (e.g bound-state of two-fermion system), and it was pointed out by C Schwartz [88, 89], much earlier than Obu, that SVM is applicable (see also [90]) for treating such types of BSP

On the other hand, Spruch and Rosenberg’s group [91-94], Percival [95], Delves [96] and Hahn [97]

have extended RRM and other related methods (see section 3.3.1.1) of BSP for investigating scattering problems Also the basic idea of the OVM employed in BSP and SP (see also [98-103]) is exactly the same

In general, most of the above-mentioned variational methods have proved to give quite satisfactory results in comparison with those obtained experimentally However, if one assumes that Schrédinger’s

constraint (or Schrédinger’s equation) prescribes the physical reality of a quantum-mechanical system,

then the precise relation between variationally calculated properties of the system, which may be referred to as the variational picture, and the physical reality leads to questions of a somewhat philosophical nature [104, 105] For example, on the one hand, many authors (see e.g [106-108, 79, 95,

101, 102, 106-108]) have insisted on the following restricted rules for distinguishing, qualitatively, between all the variational methods of BSP and SP, namely that:

i) A method of approximating the solution of Schrédinger’s equation is said to be effective if it involves a criterion which shows that one of the results is better than the other with respect to the unknown exact solution

ii) Only effective methods are able to deliver trustworthy approximations

On the other hand, it has been pointed out in the literature (see e.g [6, 9, 104, 105] and section 3.1.1.2.b), that even with effective variational methods (e.g RRM), the fact that some of the results are

better than the others does not mean that the variational picture converges to the true physical one This peculiar behaviour of variational pictures has stimulated the investigation of more restricted criteria

Moreover, converging variational pictures obtained by effective variational methods are considered

to be unsatisfactory for expressing the physical reality of a quantum mechanical system or process The ideal variational method, which should be unified one for BSP and SP, is then thought of as one which

confines, in a monotonic way, the exact physica! arguments | in regions of arbitrary high accuracy

relationship between various VM, especially those of SP, and the possibility of expressing most, or all,

O1: *

DB ne nified manre G he fh D FT C pS D "ried as O VU D i CALL = Ũ "PP" Q

of the relation | between the CVM and SVM Later on, Kato 0 [106 followed by! Kahan and Rideau [ 110]

functional

Oiseiwitsc and John see also , have stressed the ideas of Kato in the case o CMV and MVM Also Kanellopoulos and Wildermuth [115] have demonstrated the unified basis of the CVM Recently, the relationship between CVM and Schwinger-like methods [55, 56, 116, 117] has

been re-investigated by Garibotti [118] and Singh [119, 120] by means of a comparison with Padé-

approximants {121] Additionally, Singh and Stauffer [122] have presented a unified formulation for

these methods using the approximation theory of linear operators in Hilbert-space and the Lippmann-

Schwinger equation Very recently, however, it has been pointed out [79, 123] that the essential difference between Hulthén-like VM as well as the QVM is that they use different test-function spaces = ~

Finally, it has been shown by Wildermuth and Tang [124] that the representation ¢ of # Schrodinger’ S

bound-state and scattering problems based o on 1 the cluster model of nuclei (125, 126]

The present work is concerned with the representation of the VM of BSP and SP using the

168 M.A, Abdel-Raouf, On the variational methods for bound-state and scattering problems

projection formalism The unified basis of these methods is discussed in details Also there is presented a comparative study between them as well as a short review of their applications, especially in atomic physics

2 Generalized variational and projection formalism

In non-relativistic time-independent quantum mechanics, Schrédinger’s equation is equivalent in form to the conventional eigenvalue problem:

or

where E£ and H are the total energy and Hamiltonian, respectively, of a quantum mechanical system

described by the vector |) (The boundary conditions of |) characterize various quantum mechanical

systems, e.g bound-state system, scattering process, etc These boundary conditions are given in the following sections according to the considered problem.)

Now, the Schrédinger constraint can be re-stated according to eq (2) as follows; a true physical system or process described by the observable (H — E) is well expressed, microscopically, by the

expansion space |) if and only if Schrédinger’s vector (H — E)|#) defines a null space It is apparent

that eq (2) can be written as (the projection formalism; Wildermuth [127-128Ì]):

The S, sign means summation and integration over the discrete and continuous parts, respectively, of

he expansion space The se i omposed of linez independent, b not necessarily o nogond vectors which satisfy the same boundary conditions as |)

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 169

The variation in |) is given by

|ôứ) = S ôa„ |) (6)

k

Substitution from eqs (Sb) and (6) into eq (3) yields:

where the index k runs over all possible states of the considered system

For practical calculations, |) is replaced by a trial expansion space |{) defined by

The advantage of using eq (11), which is in practice equivalent to eqs (9) and (10), for estimating the

properties of a certain quantum mechanical system lies j in the fact that or is free of any r restrictions

describing arbitrary behaviour of the quantum mechanical system, “However the weak points of of ate

nple to identify {73=75] Having assumed tha al expansion space tw) of a bound-state syste

is composed of n orthogonal vectors, then eq (11) vields n bound-states (i.e n values for E) Some of

these states are characteristic to the Hamiltonian H and the others are unphysical (In fact the situation

is much worse than that For real systems other than one-electron ones, we get n pseudo-states, some of

which are good approximation to real physical states and some are insignificant.) Also if [p{”) is

considered as the superposition of vectors describing the asymptotic and interaction regions of a

scattering problem, then eq (11) may lead to the appearance of spurious singularities (see also [76])

Equation (11) demands also that the projections of Schrédinger’s vector (H — E)|w°) into ac

{|.)} are zero For the generalization of this idea we rewrite (11) as follows

where |ớ.) is called the test-function space of the vector (H — E)}W), [79, 123}, and can be chosen to

clude-compone D e superposition of compone 0 al expansion space Indeed

projection formalism given by eqs (9), (11) and (12) follows directly from the variational principle (10),

170 M.A Abdel-Raouf On the variational methods for bound-state and scattering problems and consequently, any change in the terms of these equations follows from a change in the basic

principle For example the fact that the projections in (12) are zero is due to the restriction that |¢,) is dimensionally equivalent to |6y,) of (10), which is based on (9) Attempts at suppressing this restriction implies a restatement of (10) In other words if we choose |@,) to be arbitrary large, we get

helps us avoiding the appearance of unphysical phenomena characteristic of eqs (9) and (11), and also provides us with more flexibility in the treatment of variational methods of bound-state and scattering problems

Assuming that the variational pictures obtained by (12) and (13a, b) are free of any unphysical

phenomena (spurious singularities or insignificant pseudo-states), the argument that these pictures (or

any other pictures obtained by VM based on eq (10)) are approximations to the physical reality, say of

a bound-state system, expressed by eq (2) is obscure (see e.g [5] p 176, [6] p 5 and chapt V and [9, 105, 129]) Formulations which combine both pictures in one equation are called in the literature

‘quantum identities” (see e.g [106, 43, 91, 130-132]) and are shown to be as complicated as Schrodinger’s

equation [133] (A discussion of these identities will be presented later on in this work.) Nevertheless, it is

necessary now to give a precise prescription of the role of the variational theory in quantum physics As it is

mentioned above, a verification of Schrédinger’s constraint requires the exact knowledge of the terms E, H

and |) In practice, however, |) is always unknown, the parameter E is not given for certain type of

:

a h D awe a 2 a aiff 2 ao problems (58 and ntn ompie orm of A may de alin 0 con Nn advan OD anationad

treatment of a physical problem, on the other hand, the situation is handled in the following way: A trial

Dĩ pace p54 Ci€ Ca Wil it Geume a HYpoeme adi Pity t1 CHA 0 DEFOCE A O ated

AVA n

a s a

an operator O, in such a way that |y,) and O contain, in principle, all available in formation about |) and

H=PE espectively (The syste ypothetica at Olứ:t" 1s not necessarily-zero:

Furthermore, both |‘) and O fulfil a variational constraint and leads to a variational picture of the true

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 171

The arbitrariness of choosing the variational constraint allows us to generalize (14) to the form

The test-function space |¢,) can be constructed in a similar way as those of eqs (13a) and (16), from vectors (or superposition of vectors) involved in the trial expansion space |{) An essential point in

the variational theory is then the choice of the superpositions of |y{) given at eq (8) The rule is to

select them from vectors obeying quantum mechanical constraints (like the Pauli exclusion principle, symmetrization and antisymmetrization laws, etc.), inherent to the appropriate field of physics, as well

as some physical intuitions Generally, the superpositions chosen for a bound-state system are similar and dependent on the model of the system considered Those of a scattering process, however, are much more complicated, since they describe the asymptotic forms of scattered systems, their relative motions, as well as their interaction region Later on, we will discuss explicit forms of trial expansion spaces characteristic to BSP and SP of atomic, molecular and nuclear physics In the latter we will restrict ourselves to BSP and SP described through the cluster model

3 Variational methods for bound-state problems

The quantum theory [134] distinguishes between two kinds of observables associated with quantum mechanical systems; namely dominating and non-dominating observables (e.g total Hamiltonian, total

angular momentum, etc.) lead to eigenvalue problems, i.e

(Remark: Equations (19) and (20) indicate that both € and (Oy) are calculable if and only if (ply) exists,

i.e |) is a vector in a Hilbert-space.) If Op = H (i.e the total Hamiltonian of the quantum mechanical

system), then eq (18) leads to Schrédinger’s equation (1) and the eigenvalues of H, obtained by (19), are

exa the energy le or bound states) of the em The corresponding eigenvectors describe

uniquely (if degeneracies are ignored), the system in these states and, consequently, satisfy the following

boundary conditions 1 34]: tu KvViebitkvltio I+v 3

u(r) (Fy ~ 0 0 r>

Ú(r)~r”' — r0

172 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

where r and / specify the generalized position vector and angular momenta of the system Following the terminology of section 2, the variational methods of solving the eigenvalue problem (18), for Op = H, yield variational energies and variational pictures for the quantum mechanical system Also those VM which are concerned with eq (20) provide us with an estimate of (Ox) We expect that the first type of methods can be expressed using the formalism of section 2, while the other requires different techniques

3.1 Variational methods for eigenvalue problems

As we mentioned in the introduction, there are two groups of VM relevant to the solution of Schrédinger’s equation when bound-state problems are concerned The first group (Rayleigh-Ritz’ type

VM) contains observables to first order The second group (quadratic VM: QVM) involves observables

to second order In sections 3.1.1 and 3.1.2, we discuss, respectively, both types of methods for arbitrary eigenvalue problems

Or

is valid, then eq (23) is the Schrodinger constraint of the system Equation (22) describes an eigenvalue

, n forms of H, and H;, such that

By = (| Hal de) (e| Hole) (24)

is true If {lý,)} C Du, then we distinguish the following two cases

1) {lứ,)} are orthonormalized with respect to H;, i.e

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 173

1) {lứ„)} are orthonormalized w.r.t Hị, ie

in which case the F,,’s are called H,-energies The symbol ô„„: 1s the usual Kroneker-delta

From the Hermiticity of H, and H, one can easily show that:

a) the eigenvalues of (22) are real;

b) the eigenvectors |,)’s corresponding to different eigenvalues (once the degeneracy is removed)

are orthonormal w.r.t H, i.e

(x|H›|lửx) = Ôyy: for k# k’ (27)

if the E,,’s are H,-energies and vice versa (For this reason the problem (22) is sometimes referred to in

the literature as “conditional eigenvalue problem’’.)

The special case of H> of (21) and (22) being a unitary operator, leads to the conventional eigenvalue

problem

which possesses, according to eqs (24), the eigenvalues

Equations (28) and (29) imply a one-to-one correspondence between the E,’s and |y,)’s once the

; acies have | LC tl jer the E.’s such thai

E, <E, for k = 1

—K aK + I awe LÀ 2 ˆ 2 (30) Py

The direct variational treatment of the eigenvalue problem (28) was first developed by Rayleigh [23, 24],

k =1,2, ,n The RRM can be formulated in the terminology of the last section as follows:

1) The variational principle (17) is used with O=H,-FE and equivalent trial expansion and test-function spaces, 1.e

174 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

Consequently, eq (17) can be written as

where the test-function space |@,) is composed of the n Hilbert-space vectors {|y;)}7-1 The vectors |wWtz’)

obtained by eqs (35), (34) and (32), prescribe the variational pictures of the quantum mechanical system

dominated by eq (28) The eigenvalues E,,.’s represent the variational energy levels of the system

Rayleigh [23, 24] and Ritz [25, 26] proved the important relation between E,,, and EF, (the first exact energy level of the system), namely that

us), the first variational energy is an upper bound to the exact

(WIND) = Bice, for k,k’=1,2, ,n

M.A, Abdel-Raouf, On the variational methods for bound-state and scattering problems 175

is fulfilled, then we get

E,<E,, fork =1,2, ,n (40b)

McDonald [27], on the other hand, demonstrated that, if the trial expansion space, and consequently

|#;), is enlarged by exactly one component such that (g*Đ|g#?)=ôw — fork,k'=1,2, ,n+1 (41)

then the following successive relations are always valid:

Envi1S Enis Envi2S*** SE nsin S Ean SE nsinet (40c)

E, SE„.tvS E„SEu-ivS' SE (401)

Equations (32), (34) and (39) can be used to obtain variational pictures and variational energies for systems dominated by eq (28), if one uses test-function space |¢,) composed of elements which are not

involved in |‘) but rather in Dy, However, the resultant variational energies, which may be better in

quality than those of RRM, are not necessarily upper bounds for the correct eigenvalues of (28)

The generalized eigenvalue problem (see also [13] section 4.14) presented previously in eq (22) or

(23) has been first variationally treated by Ritz [25], Bubnov 1913 and Galerkin 1915 using trial expansion

spaces of the form (32) (The works of I.G Bubnov and B.G Galerkin are extensively discussed in [6] pp

10-13, sections 29, 34 and 73-82 The VM is referred to in some literature [6, 135] as Bubnov—Galerkin’s

method In other literature [136, 137] it is considered as Ritz—-Galerkin’s VM In the present work the first

indication is used.) Following Michlin [6] and Gould [138], we assume that H, and H> are Hermitian and

positive definite operators The variational principle (17) can then be written, for O = H,— EH, and

lb) =lb) into the form:

The equations possess nonzero coefficients c;,’s if and only if

2 H%)- EH H%- E„H$ sae H$- EH?

A

nk —

Hit Ex 1 2 Hàn EuH% Hữa- EHin1 2

176 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

is true The eigenvalue E,,,’s, of eq (44) are given by

They represent with |{)’s the variational energies and variational pictures of the system (22) and one

can show that, if both E, of eq (24) and E,, are of the same type (i.e either H,-energy or H2-energy) then

where |ø.) can be freely chosen from the domain of H; (or H;) A necessary condition for the fulñlment

of (46) is that |ø.) and |ýt"” are to be equivalent

Bidenharm [29] and Robinson and Epstein [139] pointed out that, if H; in eq (22) is positive definite,

then variational energies and variational pictures for this system can be determined from the system (28)

if one replaces H, by the operator H3"”H,H3"”, and uses RRM with trial expansion spaces of the form |p) = H47|w), where |‘) is defined by (32) Furthermore, the nonpositive definite Hermitian

H, implies different variational treatment of the eigenvalue problem (22) (H; is assumed to be positive definite) In this case, one may transform eq (22) to the reciprocal eigenvalue problem of Courant and

Hilbert ((5] chapt 3), which is equivalent to the so-called conjugate eigenvalue problem (see e.g

Walmsley and Coulson [140], Walmsley [141], Joseph [142]), i.e to the problem

where the é,, satisfy relations (40a)—(40f), if ¢ is H2-energy

If « in (47) is Hi-energy, then e,, satisfy the inverse of the inequalities (40a)—(40f)

Now assuming that H, is indefinite [134], then

ap PL Halas e|

lent = = Ge 1, mm)

and the inequalities

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 177

are valid It is obvious from (40g) that for indefinite H2, the Rayleigh—Ritz relation between the

variational energies and the exact ones does not hold

3.1.1.1 Applications of Rayleigh—Ritz’ type VM Mainly because of its conceptual simplicities and its guarantee of a direct relation between variational and exact energies, the Rayleigh—Ritz form of the

variational theory presented in section 2 has been widely used in various areas of physics However, as mentioned before, to different fields of physics correspond different explicit choices of trial expansion spaces In the next three subsections we will confine ourselves to some explicit forms of these spaces characteristic to systems encountered in atomic, molecular and nuclear physics

3.1.1.1a Two- and many-electron atomic systems The two-electron atoms, the so-called ““helium- isoelectronic sequence”, are described by Hamiltonians of the form (a.u.):

H = -(Vit+ V3)- Zn - Z/nt Une (51)

where r, r2 are the position vectors of two electrons relative to an infinitely heavy nucleus of charge Z

and riz is the relative distance between them The first attempt to calculate variational energies for He

and Li* atoms was carried out by Kellner 1927 [144, 145], using four superpositions for constructing

|“) at eq (32) of the form:

Xi = Atz Em (1) &, (72) Pi(Cos 612) (52a)

L„(r) and P;(cos 6,2) are the first derivative of the mth Laguerre-polynomial and the /th Legendre

function, respectively, 012 is the angle between r, and rz Aj, is the antisymmetrization operator

Later on, Hylleraas [146-148] improved on Kellner’s variational energy for the ground state of the

He atom by introducing 12 of the following ly,)’s:

alc KiTOW C CTALUTE aS Myce ol VCCLOIS, C UICCãS Pi, Gi aANQ U; are Cve OSC OD Sd V

the relation p; + q; + v; = J, where I = 1,2, etc These vectors form a complete and analytic set [105]

178 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

Table 1

The convergence of the first (singlet ground-state) variational energies (—E,1) of helium-isoelectronic sequence calculated using Hylleraas’ vectors

(eqs (53a, b))

The convergence of higher variational energies (-E,,.) of S-states of He and Li* calculated

using Hylleraas’ vectors (eqs (53a, b))

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 179

- 2.90370 -

Fig L Variation of E„¡ of He atom with @ using 18 Coolidge-James’ vectors (eq (54)) with a4; = @, @2; = 1— a, [164]

Hylleraas’ trial expansion spaces (53a) have been employed by different groups of authors (see Chandrasekher et al [149, 150], Hart and Herzberg [151], Sucher and Foley [152] and Pekeris and

collaborators [153-160]) for computing variational energies (E,,,’s) and variational pictures (|¢{)’s) for

all elements of the He-isoelectronic sequence By enlarging the trial expansion spaces (up to n = 2300, [156, 159]) and adding mass-polarization and relativistic corrections, the accuracy of the results has

steadily increased over the years Traub and Foley {161, 162], on the other hand, improved Hylleraas’

where a1; and @2; are free parameters and p;, đ; and v; are the same integers as in eq (3a) Better

estimates E,, (see Perkir D4— 100] and AdDdel-Kaou 67]), comparative to those of Peke

same n, have been carried out by varying the nonlinear parameters a1; and aj

Further improvement can be obtained by introducing a factor exp(— y;r:a) in the right-hand-side of eqs

(53a) and (54) (see Esquirel and De Téramond [168] and Radi [169], respectively)

180 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

Although, Hylleraas’ vectors (53a) have shown to be relatively effective for treating two-electron

systems, Bartlett et al [170] had doubts about the convergence of the resultant variational energies For accelerating this convergence Bartlett [171] and Fock [172] and Hylleraas and Midtdal [173] inserted components of logarithmic form (log s) into eq (53a) Nevertheless, H.M Schwartz [174, 175] and C

Schwartz [176] insisted on the value of using Hylleraas’ vectors with fractional powers, i.e fractional p,,

q; and v; in (53a) Their variational energies of the He-atom are of the same order of Pekeris’ ones and calculated using much smaller trial expansion spaces

Kinoshita [177, 178], on the other hand, emphasized the use of the generalized vectors:

where s and u take negative and positive powers From eqs (53a) and (55) we remark that Hylleraas’

expansion space is a subspace of Kinoshita’s corresponding to the case p; =q; =v; 20 for all i>0

However, it was indicated previously (see Bartlett [171] and Fock [172]) that a complete Kinoshita’s

trial expansion space is nonanalytic (An explicit proof of this fact was given recently by Morgan

{179, 180] Further discussions are also presented by Scherr [181].) However, a set of the first 80 quadratically integrable elements of (55) led to fairly good variational energy for two-electron atoms [177, 178, 182]

Nevertheless, several authors (e.g [183]) objected to the use of the correlation vectors presented at eqs (53a), (54) and (55), when considering atomic and molecular systems

The authors raised the following two critical points:

i) It is hard to give a satisfactory physical meaning for these vectors, and ii) It is difficult to deal with them in many-electron systems

Table 5

fractional powers as well as the results of Kinoshita

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 181

In order to overcome this problem, Green et al [184, 185] replaced these vectors by

x¡ = €f?Ứn, ra) P£?(cos 412) (56)

where |é) are normalized r,- and r.-vectors and P$” are the usual normalized Legendre functions

Thus |p”) is normalized in the configuration space of the system Consequently, the resultant E,,’s are

denoted as the variational energies of “‘Configuration-Interaction”, CI Indeed, the Cl-method (also

called the method of “Superposition of Configurations” {186]), was first employed, for the He-atom, by Hylleraas [187, 188] who suggested the trial expansion space:

r= @) `” > P,(cos 612) S CụXu(aeri)Xu(ar2) (57a)

I ij

where

Xxu(œr) = (2a) ”^{Q + !+ ĐI 3240 — I— 1)!f?Qar)! e-L 742 ,2ar) (57b) and L147, are associated Laguerre polynomials

Hylleraas’ results, for singlet states of the He atom, using eqs (57a, b) were less successful compared

to those obtained by eq (53a) and consequently, the development of such trial expansion spaces was

ignored for a while in the literature However, quite satisfactory results for two-electron atoms have been, later on, calculated by many authors (see e.g [189-199]) using up to 80 configurations [197] (For the Problem of varying @ in eq (57a) see e.g [200].) Nevertheless, L6wdin and Shull [189] indicated

that, because of the arbitrariness of choosing the vectors in eq (57a), (the only condition demanded is

that they form a complete set), it is not easy to give a simple physical interpretation either for the

associated coefficients (c,’s) nor for the whole trial expansion space In order to deal with this problem,

Léwdin and Shull [189] emphasized the use of trial expansion spaces composed of the so-called

“Natural-Spin Orbital” (NSO) vectors These spaces are to diagonalize th t-order ity matri

f y(rilr\)=N | t (i, Fo) We" (11, F2) dre (58)

where r; = (r;, 8;) is the space-spin coordinate of the ith electron The calculations, for singlet He-like

182 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

Table 6 Convergence of the 2°S state of He determined by increasing the number of configurations of the form (57b), [197]

variational 2174263 21751699 21752177 21752242 2.1752255 217525 2.17522%

and the natural spin orbitals are then p;a and pjb

Further manipulations using eqs (60b)—(60d) and (60a) leaves us with

Accordingly, the trial expansion space |w) is constructed from the first n natural spin-orbitals given at

OUT ) JWU ATIC v 9 ad VOCATECC d S proceau’re tedds CoO TIPTOVEMCH 5 Te Vartrationd

energies of the lowest singlet and triplet states of the He atom They have also clarified the relationship

between their procedure and Hartree-Fock [201-205] and extended Hartree-Fock [206-207] tri —- expansion spaces (see also {13] section 4.11) (For the application of these spaces in two- and

many-electron atomic systems see e.g [208-215] Extensive discussions are also given by Fischer {216)})

Finally, several authors avoided the use of polynomials in r;, r and r,2 in the trial expansion spaces

The variation of —E,,,; with the number of NSO configurations [192]

— n1 2.87860 2.89958 2.90116 2.90123

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 183

of two-electron atoms For example, Wang and Weinhold [217] chose |y;) of eq (32) to be of the form

where {|£;(r))} is a set of hydrogen-like eigenvectors Furthermore, Power and Somorjai [218] introduced

the following form:

Calculations using equations (62) and (63) showed fairly good convergence of the variational energies compared to that obtained by Hylleraas vectors (53a)

The Hamiltonian of an N-electron atom is given neglecting spin-orbit coupling and other small effects

by

m-Š{ tr + ()} 60

where r; is the position vector of the ith electron relative to an infinitively heavy nucleus of charge Z

and the r,’s represent the interelectronic distances between the N electrons Now, since Hylleraas’ and

Kinoshita’s vectors (eqs (53a) and (55), respectively) are not easy to formulate for many-electron atoms,

we restrict ourselves here to the vectors of Coolidge and James (eq (54)) In this case the superpositions

of eq (32) can be abbreviated by

jm>l

where the operator A is a many-electron antisymmetrizer, the p’s and s’s are integers and @’s and y’s

are free PA For pure configuration spaces, one ‘Puts y=0 and expands th the Yn vectors in terms

the cosines can be expanded using ‘the rules of spherical trigonometry, Moreover, one can introduce in in

q- (65)-a-scaling parameter-a—by-repiacing +; _by-ar;_ The best -value-of-a—corresponds to-a-ce tr:

expansion space that can be obtained by using the virial theory of Lowdin [200] Also, the generaliza-

1O O E - and VOS-pro COUTE TOIT WO- [TO Mdny-ele ON alOIT d raight-forwa ra proce

[183, 189] The choice N=3 and Z=3,4, at eq (64) corresponds to the so-called “lithium

isoelectronic sequence’ Ihe most usual trial expansion spaces, which involve correlation terms, of

these systems can be obtained from eq (65) by putting y = 0 Thus we get:

= Ari"r?*r$" eXp{(œun † đa; + œar:))r12 r3 r1" (66)

Equation (66) has been employed by several authors (e.g James and Coolidge [219, 220], Burke [221], Ohrn and Nordling [222], Larsson [223], Larsson and Burke [224] and Perkins (225, 226]) for determin-

_ing variational energies and variational pictures for the*S state of the Li atom using up to 60 superpositions

Also an estimate for the “S state of this atom is given by Larsson [227] using 57-superpositions Similarly the

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

Table 8 Convergence of the variational energies of the Li atom (-E,,) using Coolidge—James’ vectors (eq (66))

Cam + hat E fel Fo ng Lita 4 1; 1

Conipal ison Octweei —L, 7 Of tine Li atom catcutated by Uillerent autnors

James + Coolidge [219] 10 7.47610 Coolidge—James (eq (60))

Weiss [230] 45 7.47710 Superposition of configuration (CT)

Burke [221] 13 7.47795 Coolidge—James (eq (60))

Fischer [215] 18 7.43272 Hartree-Fock

7.47678 ì Evtended Hart Back with

747690 SACHIN ETAT TOUR WITT

7.47703 j various configurations

Table 10 Variational _hyperfine states of Li and CT atoms [229] calculated using

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 185

67-vectors, respectively On the other hand, successful Cl-calculations for 3-electron atoms were first carried out by Weiss [230,231] with 45-configurations Sim and Hagstrom [232], indicated that on employing 150 terms of mixed CI- and Coolidge—-James'’ vectors, one gets effective estimates for the *S and

*P states of Li Additional calculations for the ground-state of the Li-atom are also available using NSO- as

well as Hartree-Fock-procedures (see e.g [215]) Finally, the Be-isoelectronic sequence has been treated in

the same way as several authors [233-237] using different trial expansion spaces

All of the above mentioned calculations emphasize strongly the necessity of considering correlation

components into trial expansion spaces of two- and many-electron atomic systems However, the

variational theory is still suffering from the lack of mathematical techniques which simplify the use of these vectors especially in complicated systems Also the theory does not provide a satisfactory physical

interpretation for the effect of these components However, many authors attribute their importance to

the nature of two-body forces characteristic to these systems These Coulomb-type forces (see eq (64))

are of long range behaviour and allow for polarization effects which can be described by the correlation

vectors Nevertheless Rayleigh—Ritz’ type VM are shown to be useful for determining variational energies and variational pictures for autoionization states in the framework of Feshbach’s projection

formalism [238-243] (these are quasi discrete states appearing in e—atom scattering and atomic systems) However, the fact that the calculated eigenvalues are complex, which implies that the

corresponding operator is not self-adjoint, violates the possibility of determining upper bounds on the true eigenvalues using RRM Actually, various opinions have been mentioned in the literature about the

existence of such bounds and the importance of using other types of variational methods for dealing with

such problems (More precise discussions are given by Sharma [244]; see also Sharma and Sri Rankanathan

[245] and the references therein.)

3.1.1.1.b Diatomic molecules The total Hamiltonian H of a molecule composed of two atoms a and b

with N, and N, (N = N,+ N,) electrons, respectively, is given (in atomic units) by

ai Yer are the position vectors of the 7th electron relative to the infinitely heavy nuclet of charge

Z2 and Z2, respectively The vectors r,; and R., specify the interior electronic and nuclear distances of

the molecule (The Hamiltonian (67) is free of all spin-orbit coupling effects.) In fact the difficulty of

Carrying out a precise variational treatment for molecular systems lies in the large number of the

associated electrons and the existence of more than one center of singularity in the configuration space

(i.e more than one nucleus) For these reasons most variational pictures and variational energies given

in the literature are obtained (especially for molecules larger than H;) in the framework of Hartree—

Fock and multiconfiguration Hartree-Fock methods (Detailed study of these methods and_their

applications to various diatomic molecules is presented by Hurley [246].) However, Clary [247] has

dis SSeG ne Sse 0 r1a eC—FOCKangG ƠOnecFra Ong A 1On-1n erac ion ecnnidgue Pspe-

>

cially when diatomic molecules are investigated The author extended Previous trials of employing

= Wi Cl L1 Ð = CY 7 a VO PIS An > di Mla 9n mole wv li AA Cl ne ol ny ` OO roo si AR ° G nc Ww Alf TU AQ

order to determine variational energies and variational pictures for arbitrary many-electron ones

` ing R ciph Rate’ mothxa¬d ory A can ¬ " ecsrmÌla =¬~¬ n ^^ ae Ae a a Aan ¬

cd Od, a DFOPOscecd a Ud d 5 U Ula Wild U{) UU U Ud VJ

with ‘the complicated integrals involved His trial expansion space is developed from the r,;, components,

A

the so-called confocal elliptic vectors (see also [13] section 4.12) p*, p~ and the azimuthal angle 8,

186 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

where ơ, P„, P„¿, ŠS;, Š; and g; are integers, a; and Ø„ are free parameters In the practical use of

(69), Clary put »; = 0,1, Sj; = $j =0 and g; = 0 His variational energies of the states ‘3; and *3% of

He, and He3, respectively, are obtained by employing up to 34 dimensional trial expansion spaces The results are of very good accuracy in comparison with those evaluated with CI approaches

In all of the above mentioned applications of Rayleigh—Ritz’ type VM, one has employed trial

expansion spaces |y{) which consist of continuous vectors In fact this condition is proved to be

essential for the validity of RRM [250] On the other hand, the investigation of some diatomic molecules (e.g H3) has indicated the importance of involving discontinuous vectors (i.e vectors which

fall of to zero at definite regions of the configuration space and consequently have discontinuous

derivatives) into the trial expansion space (see e.g Hall and collaborators [251-253]) This implies the conversion of the Rayleigh—Ritz eigenvalue problem (eq (28)) associated with the considered system into a

Table 11 Variational calculation of the '33 state of He2 and the

?š* state of He} [247] using Hylieraas’ vector

2 4.75919 4.913889 Comparison between variational calculations {247} of

4 4.77076 4.939621 the 2X} state of He} obtained by different forms of |x;);

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

Bubnov-Galerkin eigenvalue problem The procedure is as follows:

Equation (28) is rewritten as

(T+ Vb) = Ely),

187

(70a)

where T and V are the kinetic and potential energy operators, respectively A rearrangement of the

operators in (70a) yields

where yw is a Lagrangian multiplier and G is Green’s operator (E — T)"* Equation (70d) coincides with

the Bubnov-Galerkin's eigenvalue problem (22) at H, = VGV and H,= V On defining an appropriate

trial expansion space |‘), the eigenvalues of (70d) can be obtained by Bubnov—Galerkin’s variational

method using the form (eq (45))

where V has to be definite and Hermitian [139] The operators VGV and V in eq (71) are not of

differentia ne_and consequen > ma

Fig 2 Eigenvalues (ø„¿) obtained from eq (71) and the correspond-

ing energies for a central field problem [143] with a Fermi-Thomas

potential

Fig 3 Eigenvalues (4,.) obtained from eq (71) for a central field

problem [143] with a Herman-Skilman potential.

188 M.A Abdel-Raouf, On the uariational methods for bound-state and scattering problems

can be ordered such that

which yields an upper bound for each E,

3.1.1.1.c Nuclear systems Contrary to the case of atomic and molecular systems, the Coulomb- forces of nuclear systems are relatively unimportant and it is known that they are rather dominated by forces of short-range character However, the precise mathematical form of these forces, and con- sequently of the total Hamiltonian, is still unknown and one is often obliged to add phenomenological

interactions for improving the variational pictures Also, various opinions (see Wildermuth [254]) about

the arrangement of nucleons inside the nucleus led to the development of different nuclear models (e.g

the harmonic-oscillator shell model [255,256], the collective model [257] and the cluster model

[258-261]) and to an unlimited number of choices of the trial expansion spaces Accordingly, the application of the variational theory to nuclear systems becomes not only necessary for obtaining

variational energies and variational pictures for these systems but also for testing the quality of the

nuclear model under consideration Preliminary variational calculations, for example, can be obtained

for any nuclear system using the simple harmonic-oscillator shell model In this case the superpositions belonging to |y) (eq (32)) are chosen from the eigenvectors of a harmonic oscillator (e~°®’) which are

further simplified especially when complicated nuclei are investigated Improved variational pictures

and variational energies of many nuclea em Đe e.0 62—26 and [260] and the reference

therein) have been established by employing the cluster representations of nuclei Indeed the fact that

mole P OMmMpaeseG GO Hmoie e An GO atom "nrera ing i-Cacn- OMer, Hoge CO O-man

authors [124] that the nucleons of a nucleus may tend to describe various substructures (grouping of

^ cÝ ¬~eG 4 Thạc ` = he ^hoxi^e = 4 a 2 2 a 499 4 28 era fae L 9 2

corresponding to different Clusters The resultant variational picture is called the resonating group

model In fact the original form of the resonating group theory of Wheeler was basically refined by

Wildermuth-Kanellopoulos and collaborators (see the above mentioned references) These authors

restricted the substructures to fulfil in addition to various quantum mechanical constraints, the Pauli

exclusion principle ({124] chapter 3) On the other hand, Wildermuth and Tang [124] used the

variational theory and the cluster model for a unified treatment of nuclear bound-state and scattering problems Their variational procedure (eq (9)) in case of BSP is exactly RRM (see also Wildermuth and

McClure [266] section II-section IV) We now try to deal with this procedure in the framework of the

terminology given in section 2 We proceed as follows: Consider a nuclear bound-state system of N nucleons We propose an observable M1, instead of the unknown H, which exactly dominates the

harmonic oscillator eigenvalue problem) Accordingly we take the eigenspace of W, to be the trial

M.A, Abdel-Raouf, On the variational methods for bound-state and scattering problems 189

expansion space |) Both W, and |~{”) are reformulated by physical intuition following quantum

mechanical rules The final form of W, and |‘) are used to obtain the variational pictures and the

variational energies of the N-nucleon system (Remark: A slightly different procedure was proposed,

later on, by Drachman [267, 268] for dealing with atomic scattering processes with targets of uncalcul- able exact expansion spaces (e.g two- and many-electron atoms) The method is referred to in that literature as the theory of model.)

For illustrating the above technique, let us start with W, which defines an N-particle oscillator, i.e

_ 1 N 2 Mo? & 2

where p; and 7; are the momentum and position vectors respectively of a particle of mass M oscillating

with angular frequency w Now according to the cluster model, some of the N nucleons may form

several clusters and consequently, if there are » groups, we get

Daman, : (74)

where pz, is the number of nucleons in the ith cluster

Assuming that R; and P; are the center-of-mass coordinates and momenta, respectively, of this cluster, such that

where H, depends on the relative coordinates, and their derivatives, of the ith cluster Now the whole

expansion space of W, given at eq (73) is the eigenspace of the N-particle harmonic-oscillator and has the

form:

190 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

and |&,;) is a vector describing a one particle oscillator multiplied by the appropriate spin and isobaric

spin The vectors {|y;)} form, as we know, a complete set

Now in order to write down the explicit forms of W; and the corresponding trial expansion space of

an N-nucleon system we proceed, similarly to Wildermuth and Kanellopoulos [258-260], by considering

as an example the a-cluster model of “Be In this case we have two clusters of center-of-mass coordinates R, and R2 given by

The two clusters have the relative coordinate R = R; - R; and the whole system has the center-of-mass coordinate R, = 3(Ri+ R2) The relative coordinates p; in each cluster can be expressed by:

pi =i — R;

where j = 1 at i= 1, 2, 3, 4 and j =2 at i=5, 6, 7, 8

The operator W, of eq (76) can be reduced to the form:

Wi = Hai + Ha2t Tea Pet T— 1 R; + ag P + Mo“R“, (79)

where the Hamiltonians H,, and H,2 depend on the coordinates p; of the clusters, e.g H,1 has the form

M’*o

ij=1

The operators P2 and P’ at eq (79) correspond to the conjugate momenta of the center-of-mass and the

relative motion respectively

The vectors €, and & 2 describe the two- clusters, é, designates the spins and isobaric spins of the 8

AUCICONS, €;, and éicm F€Ppresent the ive motion of the two clusters and the oscillatory motion o

center-of-mass of the whole system, respectively, The operator A is an N-nucleon antisymmetrizer

Xi = Aj exp(-1 2y ` p; | Yiv1YotT2BavaBavs |} eX P(T? 2y > 1) ysvsyerteBrvrBae}

j=l

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 191

Be 0 ot -57.1 -56.5

2! -53.8 -53.6 4* -45.5 -45.2

where y; refers to the spin-up for the jth nucleon; 8, v; and 7, are the spin-down state, isospin-up

neutron state and isospin down (proton) state, respectively, for nucleon j Y4, is the spherical harmonic,

in space angles ổ and ó with /= 4 and m =4 and y is a free parameter Now the operator W, (eas

(79, 80)) can be used with the trial expansion space given at eqs (81-83) to obtain variational pictures

and variational energies for "Be based on RRM [260] Similarly one can sol

ae ^ _ * 4 3 xa

may be treated using the same procedure [124, 262, 263]

ve for “Li [265] by using

192 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

Nevertheless, various variational pictures and variational energies for any nuclear system can be

produced if the second term on the right-hand-side of eq (73), which represents the interaction between

the N nucleons, is selected to follow other physical intuitions In spite of this fact the assumption of the

cluster model remains preserved

As we see from the above discussion the variational treatment of complicated nuclear systems using

the cluster model requires first the determination of effective variational pictures and variational energies of two-, three- and four-nucleon systems, i.e deuteron, tritium and a@-particle, respectively

Such pictures have been well developed using RRM and the Monte-Carlo numerical procedure by Schmid and others [269-272]

3.1.1.2 On the variational energies and variational pictures of Rayleigh—Ritz’ type VM In section 3.1.1 we

ed that RRM and Bubnov—Galerkin’s VM lead to F's and lu?) for bound-state sys

by the following equations, respectively

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 193

Also for all k’s of the same symmetry, the elements of the sets {F,} and {E,,,} satisfy the following inequality (H, and H; are assumed to be positive definite and Hermitian):

The upper bounds can be successively improved by enlarging the trial expansion space In fact this

rigorous relation between the variational and exact energies emphasizes investigating the convergence-

concept in Rayleigh—Ritz’ type VM We distinguish the following two kinds of convergence (see also

[6}):

a) The energy-convergence (it is important to mention that successive improvement of E,, does not

suppress the convergence requirement):

pra ©€ eding = ne why Ve = no wo 1a C eds c© ‘a O ana S0 imn 4 a C QO O † : = is de Ine in 2

separable Hilbert- -space y, where for any f.g © x, both (f|g) and lI= (ip? exist

i) Michlin—Bonitz’ criterion: Michi n 6] Satz 1 Tp 70 and Satz Tự PP 312-31) 4 and Bonitz ((273] Se Satz

LJÊ LÌ `2 J ¥ ° wv O O OW alidC O Cl

energies, respectively, of RRM:

194 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

is valid for all k’s of the same symmetry, if the set of vectors {H,|y,)}7~, is complete in y (Notice that a set of vectors {|é)}7_, C y is said to be y-complete if there is a vector |€) C y such that lim; ||é, — é|| = 0.)

The \ sign in (90) indicates that E,, 2,41, for n=1,2, ,%, ie the convergence is from above

In order to compare with Kato’s criterion [271], we split H, of (86a) into its components, i.e

For atomic and molecular systems, V is well defined and satisfies the so-called T-boundedness [8, 274]

Thus, if D7 and D;2 are the domains of the Hermitian operators T and T'”, respectively, then there are

constants C;, Cz, C; and C, such that

IV/ll< Cfl|+ CITA for all ƒ€ Dr (92a)

and

are valid Consequently if Dc+,;, is the domain of the operator C+ H,, C>0, then Dc 4, = Dr

Now we restate Michlin-Bonitz’ criterion as follows (see also [104]): RRM guarantees energy- convergence if the set of vectors {ly;)}i-1 is Nc+r-complete, where Ncir is defined at (88) with

O=C+T: C is an arbitrary positive constant This is equivalent to saying that energy-convergence is

demanded if {(C + T)'ly;)}7-1, and consequently {(C + T’”)|y,)}7—1, is x complete

ii) Kato’s criterion: This criterion [274] states that: the RRM guarantees energy-convergence if

{ly }i1 is N c+ r-complete, or if {(C + 7)|x;)}7-¡ 15 x-complete

Following Klahn and Bingel [104], we introduce the following remarks on Michlin-Bonitz’ and

Kato’s criteria:

c) Kato’s criterion is not necessary since it can be replaced by the criterion of Michlin and Bonitz

d) Michlin=—Boni iterion becomes unnecessary if there is a guarantee for approximating the exac expansion space to arbitrary good accuracy

e) No criterion can be made on the acceleration of the energy-convergence without developing a criterion for the quality of the trial expansion space under consideration

Now before inserting Michlin’s criterion ([6] section 78) for the energy-convergence in Bubnov-

Galerkin’s VM, we introduce the following definitions and lemmas:

M.A, Abdel-Raouf, On the variational methods for bound-state and scattering problems 195

where E,, is defined by eq (86), if the operator Hi'H, is completely continuous in y In other words (93) is true, if one of the Hermitian operators H, and H) is positive definite, while the inverse of the other is completely continuous

3.1.1.2.b The |)-convergence With this convergence it is always meant that for all k’s

is satisied According to the previous subsection, however, the superpositions belonging to |/f# are assumed to satisfy one of three kinds of completeness; namely that {[y;)}7-¡ 1s either Nc+w¿y- OT

Nc+Hy- OF ¥-complete Thus there are three definitions of the norm || || given at (94) and consequently

three types of |y)-convergence, namely Nic+ny*>, Nec+ny- and y-convergence Also the following

lemmas are true [104]:

1) Ncc+up-convergence > N c+ ny-Cconvergence > y-convergence, i.e

wie? 4 ti llic+uy > Wee) 7 Vrllcory > We 7 Wille -

The inverse of this lemma is not valid

2) An N(c.„-convergence is guaranteed in RRM if and only if

The last two lemmas have the interpretation that RRM can be adjusted to guarantee energy-

convergence bu ot jy)-convergence (see also {6} p 35 therefore e variational energies of 2

quantum mechanical system may converge to the exact | ones, while the variational pictures are

inaccurate and Schrodinger’s vector (Hi — E„)lựP » possesses a relatively large length This is a rather

discouraging conclusion, if the purpose of RRM was to satisfy Schrodinger’s constraint approximately

(For further discussion of the convergence-concept in RRM see e.g [275-278].) However, methods of improving Rayleigh—Ritz’ variational pictures are given by several authors (e.g Biedenharn and Blatt

[29]) In this case one assumes first that |w%) and |y;,) are related by

196 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

(WE | al?) = ES re’ (98a)

From eqs (96) and (98b) we get

(pee Aili!) = Ed ae + (Ex — Ex XQ.) + O(Q’)

= Ex Sit (Ex — Ex pte | Opi) + O(Q’) (99)

For k = k' we get

Ey = (te Hilt) + O(Q*) = En + O(Q)) (100)

At k# k' eqs (99) and (100) lead to

(0§)\Olugb = SE HE) + o(@9) (101)

Equation (96) can be written as

lự.) = (+ ÓØ} lá) = (1~ O)|lgix)+ O(Q?)

or

From eqs (101) and (102) we get the following relation:

If |(w&)’s are normalized but not orthogonal, we obtain

lứ)=|##»—- ® (hte Wee) — Bu ote We +0(Q?) (103b)

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 197

lự#% = (1+ OH.)|ự,)

The condition: (W|Holw&) = H„xô«x:, leads to the following form of |„):

If this condition is not satisfied we get

lhe) = MPLA WR)! lyre)

_ « #|Hjl0t)— E„(w|H|wf#> 5

Equations (103a,b) and (104a,b) are stationary with respect to the variation of |/} Thus an improvement in |) can be obtained using an iteration procedure Other formulae for improving the lw)’s of RRM are given by John and Williams [279] with applications to the bound-states of a charged

particle moving in simple central potentials These have the disadvantage that they involve Green’s

functions

3.1.2, Quadratic Variational Methods (QVM)

Variational techniques which involve operators to second order are indicated here as quadratic variational methods Indeed, some of these methods, although of equivalent effect, are represented in

the literature under different names which are also confused with variational methods of scattering

t 1 L,fIo I— tai): j >U a H

is sufficient for establishing energy-convergence and |y)-convergence in both Bubnov—Galerkin’s VM

and RRM (if Hz is the a unit operator) However, from eqs (42), (32) and relation (105b) we notice that the latter involves a functional of basically different form and, consequently, there is no way to satisfy

condition (105b) by employing these VM Also if we put O = H,— E,,H, in (105b), we get

ang the Omnparison petween eas 4 ANG padas fo the conclusion na ne aria iong 1eort

does nof guarantee, in bound- state problems, good approximation to Schrodinger’ S equation, when the

Pmer (` h r1 ON ^ a O he AmnoanAan ^ h F1 ^vy a1 ^^ AQAA hy

`2 Ul LJ7 1ö Cl ¥ s Bs s H a ) Ja +

argument suggests variational techniques i in which the vector O|y{") is tested into spaces consisting of

198 M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems

elements (or superposition of elements) of the set {Oly;,}7_, where

n

defines, as before, the trial expansion space Consequently, we consider the special case

in which the variational theory (17) leads to the form

where p is a weighting factor which vanishes at the origin and is positive elsewhere

The eigenvalue problems (22) and (28) can now be variationally treated using QVM (108a, b, c) with

O= H,- £,42 and O= H,— £,, respectively [he latter case leaves us with

he condition (ức li ) = 1 yields £), = (ứ: 1lÚt ), where all characterIsfics o are valid

The functional in (109a) can be expanded as follows:

Fr = (Wb? |Hiplb®) — 2E, we |Hip|b) + E2(wf°|p|t") (109b)

On substituting from eq (106) into (109a, b) we get the following system of secular equations in the ¢;’s:

2; o{(w|Hin|x)T— 2Euw|Hip|x) + E2@wlp|x)}=0, — ¡=1,2, ,n (110)

j

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 199

These equations are meaningful if and only if

det{(y:|H3p|y,) — 2Ea (xi ipl; + E2Qxlply)} = 0 (111)

Furthermore eq (111) requires additional condition in order to find the variational energies and the variational pictures An adequate condition on F?, of eq (109b) is to demand that

is valid (see also [9] p 275) Thus, eqs (109b) and (112) yield

On the other hand, the secular equations (110) can be simplified to the form

(y:lo(H,- E,W) =0, ¡i=1/2, ,n, (115a)

200 M.A Abdel-Raouf On the variational methods for bound-state and scattering problems

Similar to Rayleigh—Ritz’ eigenvalue problem, we now treat Bubnov—Galerkin’s equation H,|y) =

EH,|*) by considering that the functional

Fễ = (f°|(H.— E,H.)`p(H - E„H,)|ýt")

= (Úf|\Hinlựt")— Ea{twt"'|HipHalw"))+ (ýt")ÌHpH|wt)}+ E,(ÌHap|wt"), (118)

is confined by the variational principle:

The quadratic variational methods presented above, require the evaluation of the complicated matrix

ele Aiplxi), it pix), W|HipHàÌy,) and (.|H:pHhly,) Thịs difficulty can be removed by

selecting the weighting factor, p, appropriate to the investigated bound-state problem

Suitable QVM have been formulated by several authors, mainly for dealing with Rayleigh—Ritz

eigenvalue problems These methods are discussed in the following paragraphs:

a) James—Coolidge’s QVM (or the mean-square local energy deviation method [30,31]): The

mean-square local energy deviation is defined by

M.A Abdel-Raouf, On the variational methods for bound-state and scattering problems 201

is minimized w.r.t the variational parameters involved in |y) Also the final F%, is compared with F2

obtained by Rayleigh—Ritz variational energies and variational pictures

b) Frost’s QVM (least-squares local-energy methods [{32-34]): This method starts from the obser- vation that [282], as |) is the correct solution of Schrédinger’s equation

then the relation

is true at all points of the configuration space, even though both H, and |) are functions in this space

However, in practical Rayleigh—Ritz’ calculations, the parameter

varies in the configuration space and becomes stationary with respect to any variation in the trial

expansion space, only when |y{) is a good approximation to |y) Consequently, one may define a local

energy, at the point p, by

and demand that the mean-square deviation of E„(p) from its mean value is minimum, i.e