THE PSEUDO-ATOM : A SOLUBLE MANY BODY PROBLEM

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THE PSEUDO-ATOM : A SOLUBLE MANY BODY PROBLEM

M. Moshinsky, O. Novaro, A. Calles

To cite this version:

M. Moshinsky, O. Novaro, A. Calles. THE PSEUDO-ATOM : A SOLUBLE MANY BODY PROB- LEM. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-125-C4-140. �10.1051/jphyscol:1970422�.

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JOURNAL DE PH\'SIQUF Collorlirc C4. .\icl~plbtjic,~~t aii t i o 1 1 - 12. T O I I I L J 3

1 .

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1970. pa+y C3- 125

THE PSEUDO-ATOM : A SOLUBLE MANY BODY PROBLEM (*)

M.

MOSHINSKY, 0 . N O V A R O (t) a n d A. CALLES (++) Instituto de Fisica, Universidad d e MCxico, Mtxico,

D. F.

Resume. - Si on neglige tous les effets relativistes, un atonie ou plus generalement un ion de charge Z, est un systeme de 11 fermions de spin t qui sont dans un potentiel coulombien attractif et interagissent par l'inter~iiediaire d'une force de Coulomb repulsive. Conime ce probleme n'a pas de solution exacte pour 11 > I . Lln grand nonibre de techniques d'approximation ont Cte employees dans son etude et, dans beaucoup de cas, sans qu'il soit fait une analyse critique suffi- sante. Dans cet article on remplace le potentiel coulombien conmun par un oscillateur harmonique attractif et la force d'interaction par un oscillateur harmonique repulsif. Le modele qui en resulte, que nous appelons pseudo-atone, peut Ctre resolu exactement pour n'iniporte quelle valeur de ^n.

On applique alors a ce probleme des procedes d'approxiniation bases sur les niethodes de pertur- bation, de variation, de Hartree-Fock, de la variance, de la niatrice densite, etc ... On discute, dans toutes ces methodes, la convergence vers les v a l e ~ ~ r s exactes de I'energie et de la fonction d'onde, tout d'abord pour 11 = 2. electrons, ensuite pour des pseudo-atomes a couches conipl&tes, et ceci jusqii'a I 1 2 electrons.

Abstract. - If we disregard a n y relativistic effects, an atom, or more generally an ion of charge Z, is a system of 11 fei-mions of spin whicli are in an attractive Coiilonib potential and interact through a repulsive Coulomb force. As this problem is not exactly s o l ~ ~ b l e for any 11 > 1, a large number of approximation tecliniqi~es have been used in its discussion in many cases without sufficient critical analysis. In the present paper we replace tlie Coillomb c o ~ i i ~ ~ i o n potential by an attractive harmonic oscillator and tlie interacting force by a repulsive liarriionic oscillator. The resulting model whicli we call a pseudo-atom can be solved exactly for any 11. We then proceed to apply to this problem approxi~iiation procedures such as those based on perturbation, variation, Hartree-Fock, variance, density matrix, etc. metliods. We discuss the convergence of all these methods to the exact energy and wave function, first for 11 = 2 electrons and then for closed shell pseudo-atoms that go up to 112 electrons.

1. Introduction.

- If we disregard any relativis~ic was available, led t o systematic procedures for effects, a n atom, o r more generally an ion of charge determining the correlation effects. In particular Z, is a system of ir fermions of spin ?, whose hamil- Sinnnoglu

[ 5 ] ,

among others, developed procedures tonian, in atomic units, is given by for dealing \vitIi these correlation phenomena in a perturbation o r variational way but starting with

' I

₊

1

¹¹

I 11 HF

states. Other variational procedures based on

j < j = 2

1

r j ^- r j

1 .

the variance

161,

rather than on the expectation value

AS is well known, tlie eigenvalues and eigenstates of

(1. I )

cannot be obtained exactly for any 11 > 1.

Thus from tlie very beginning of the applications of quantum mechanics t o atoms of more than one electron, use was made of approxiniation t e c h n i q ~ ~ e s whose degree of sophistication has been increasing with time. Initially one was satisfied ivitli first o r second order perturbation calculations [ I ] ^01- simple variational analysis [ 2 ] . Later tlie metllods developed

by

Hartree [3] a n d their antisymmetrized version [4]

( H a r t r e e - ~ o c k (H F ) ) came into vogue and are still much in use today. The interest in tlie residual part of t h e energy a n d wave fi~nction, once the

H F

solution

of the liamiltonian, liave been used by Conroy [7].

The density matrix [8] has also played an important role in the attacks or1 atomic problems and, in parti- cular. the concept of natural orbitals derived fro111 it by Lowdin [9] has proved both conceptually

and

numerically fi-uitt'~11.

Wliile tliese and other approxirnntion procedures have been discussed at great length, their critical analysis has lagged behind their systematic application.

Thus sometimes tlie results of tliese procedures liave questionable validity.

This situation has led us t o investigate many body problems in wliicli tlie common potential and inter- acting force liave a simple form for wliicli exact -

solutions are available. We could then apply the (*) Work supported by Comisi6n Nacional de Energia approximation techniques of atomic physics t o tliese Nuclear (Mexico).

(t) Research Associate of lnstituto Mexican0 dcl Petr61eo. many body problems and test their

re dictions

(tt) Graduate Fellow of the Ford Foundation. against the exact results. Specifically, the harmonic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970422

C4-126 M. MOSHINSKY, 0 . NOVARO AND A. CALLES

oscillator potential is one such simple interaction

and if we replace the Coulomb potential by it in (1.1)

I.

e.

we get the hamiltonian of what we shall denote as a pseudo atom

If we introduce the n - 1 relative Jacobi coordinates by the definition

and the center of mass by

the hamiltonian (1.1) takes the form

Looking at the hamiltonian (1.5) from a classical viewpoint, we see that the restoring force for each normal coordinate is proportional to -

I':

with the coefficient being Z -

¹¹

for i

=

1, ...,

¹¹ -

1) and Z for

i =

11. Thus the system will be clearly unstable for Z <

11,

and for Z

=

n we have free particle motion ill all the relative coordinates ( 1 . 4 ~ ) . Therefore we will only be interested in problems in which Z > ir. We shall denote pseudo-atoms of charge Z by tlie corresponding cliemical symbol preceeded by the letter P and with as many upper indeces + ^as

tlie degree Z -

it

of ionization. Thus, for example, P L i f is tlie pseudo-atom of charge Z

= 3 and number

of electrons

it =

2.

The energy levels of the hamiltonian (1.5) are given by

where N is tlie number of quanta associated with the states in the

1t

- 1 relative coordinates, while N' is tlie number of quanta of the state associated with the center of mass coordinate. For Z

=

2

11

tlie energy levels are plotted in figure 1 in which the energy scale is given in units of (2

TI)%.

TIie eigenstates of the hamiltonian (1.5) and of the other possible integrals of motion of the problem, are more difficult t o determine as we must also satisfy tlie Pauli principle. While the antisymmetrization of the wave function is simple in terms of the coor- dinates

vi

and their corresponding spins, it is more complicated when we introduce the normal coor-

FIG. 1. - Energy levels for the pseudo-atom characterized by the quantum numbers (N', N) in (1.6) with Z = 2 n .

dinates

v f .

I n the following sections we shall give tlie explicit eigenstates in a number of particular cases as well as sketch the procedure to be followed in general.

The simplest many body pseudo-atom is the one that has n

=

2 electrons. As many of tlie usual approximation techniques of atomic theory are already applicable in this case, we discuss it in detail in the paper both because of its intrinsic interest and the insight it provides for the 11-electron pseudo- atom.

2. The two electron pseudo-atom. -

Before we proceed to discuss tlie exact and approximate solu- tions of our problem we introduce tlie notation used in the present article for harmonic oscillator states.

The single particle liarmonic oscillator state corres- pollding to tlie hamiltonian

will be given in any one of tlie following notations 1/9,,~,~(o,

^I') =

I

^{W , 111111}

>

⁼

= o 3 I 4

~ , , , ( o " ' r ) Y ,,,, ( 0 ,

rp),

(2.2) where Y,,,, is a spherical harmonic and R,,,(r) tlie radial wave function as defined in reference [lo]. When the frequency

o =

1, we shall suppress tlie o every- where it appears, i. e.

When dealing with a two particle harmonic oscil-

lator state in which tlie first has frequency

o,,

and

the second

^02,

the state of total angular momentum L

and projection M will be given in any one of the follo-

wing notations

THE PSEUIIO-ATOIM : A SOLUBLE MANY BODY PROBLEb1 CJ- 127

where the square bracket stands for vector coupling of the angular momenta I,, I, to L. When both

o, = o2 =

1 we shall suppress thc frequencies in the

Y

and the kct.

When instcad of the coordinates r , , r,, we use the Jacobi coordinates r ; , r i for thc two particle system, we sl~all denote thc

wave

functions with primes

$', Y'

and the kets with round brackets. Also, whcn possible, the quantum numbers will be primed.

Thus for cxamplc, for

o , = to, =

1 we have Y~ililliliL~,,,.(r~, r;) = 1

^{t i ' ,} l', 1 1 ; 1; L'

M')

=

a)

EXAC-r SOLUTION OF T H E

Two ELI:CI.R~N SYS-

TEM. -

For

11 =

2 it is convenient to carry out a scale transformation in cquation (1 .3) of the form

which will give us the hamiltonian

=

(I

-

2

3;) r ; 2 ]

+ f

^[P>2

+ I.;?] , ^(2.70) where

5 = z - I . (2.76)

The advantage of this scale transformation is that in the non-interaction part of the hamiltonian the fre- quency is now

w = 1

and thus we avoid bothersome factors in any expansion of the exact solution in terms of the non-interacting states.

The eigenstates of (2.7) o f total orbital angular momentum L and projection M are given by

and their corresponding energies, which are indepen- dent of L, are

As in the case

11 =

2 the primcd coordinates are given by

it is clear that the states with even (odd) I; are symme- tric (antisymmetric) under exchange of the coordinates

r ,

and

r,.

Thus the Pauli principle requires that

:

States with even 1; are singlet

( S =

O),

(2.1 1) States with odd 1; are triplct ( S

=

I), where

S

is the total spin.

In particular, the ground state of our system has

11; =

I;

= n' =

1;

=

0 and thus is a singlet whose wave function has the form

and its energy is

Eb =

.2[(1

-

2 6)" + 1] . (2.13) We proceed to expand both

YJ;

and Eb in a power series in

(5,

which will allow us to carry an infinite ordcr perturbation analysis of our problem starting from the non-interacting states.

h)

T H E PERTURBATION APPROACH.

^-

If we consider the interaction in (2.7) as a perturbation with 6 being its parameter, the perturbation expansion rcquires the development

Eb = E o + E 1 6 + . . . + E p 6 P + . . . , ( 2 . 1 4 ~ )

'f',', =

A(Qo + 6 + ... + cPP SP +

'.'),

(2.14b) wherc the coefficient

A

is selected so as to guarantee that @, is normalized. Both Ep and

Q1,

are independent of s .

From (2.13) we immediately obtain that

The determination of @, requires a little more thought. We first notice we could expand

where [lo]

x

exp [- ⁵

1

^(I

^-

2 6)"' r2] ] Rno(r) r 2 d r

where r is a gamma function. For n

=

0, a, reduces

to the first factor in (2.17) and thus from (2.12)

we see that the coefficient A in the expansion (2.146)

is given by

C4- 128 M. MOSHINSKY, 0. NOVARO A N D A. CALLES

Replacing now (2.16)in (2.12) and using the deve-

lopment

we see that in the expansion of A-I

Yb, @,

is given by

with

n' (2 p

-

1)

!

x

--

. (2.20b)

2'-' ( p +

^n') !

(p - n')

!

Thus we have the explicit form of

@,

as a function of the coordinates v;,

u;.

Had we wanted the

@,

in terms of the original coordinates u,,

r,, we just

have to develop round bracket states in terms of angular ones, e. g.

where <

⁽

> are tlie well known harmonic oscillator transformation brackets [ l l ] . In tlie present problem we have even tlie further restriction

0; =

0, but we put the more general expansion to include also the Hartree-Fock case to be discussed below.

We are now in a position to compare the results of the exact and perturbation calculations both in regard to energies and to the overlap of the wave functions. For tliis purpose we define

We can then plot, as a f ~ ~ n c t i o n of

y ,

both 6,

-

E;,

and 1 - /

^{( y A 5}

).P) 12, (2,230, b)

Eb

(xPj xP)

where the round brackets stand for the scalar pro- ducts and

A discussion of numerical results for up to p =

10 will be given in Section i) of this chapter.

c )

THE VARIATIONAL

APPROACH.

- If we start from the non-interacting states in the variables u,, u2 of zero total orbital angular momentum, i. e.

In, I n 2 1 0 0 > (2.25)

the variational analysis i~nplies that we calculate the matrix of the hamiltonian Je of (2.7) with respect to these states and proceed to diagoiialize it to obtain the energy values and eigenstates, the latter as linear combinations of (2.25). As the matrix is infinite, we have to cut it somewhere and a reasonable way would be t o cut it at a given maximum number of quanta which we denote by

p.

The positive parity of the ground state indicates that p must be limited to even values. The set of states (2.25) we consider is then restricted by

This implies that equivalently we can work in the primed system of coordinates r ; ,

r;

where tlie corres- ponding states (2.21) will be restricted by

From (2.20) we see that all kets required in tlie present problem have

11; =

0 and thus we need only the matrix elements [12]

Diagonalizing the matrix for 2 n;, 2 M; <

^p

we obtain, among others, the energy and normalized wave function for the ground state. We can then, as in (2.23), compare them with the exact solution, which we will do in section

i ) .

It is important to note that our variational ground state, as the perturbational one of the previous subsection, is given in terms of tlie primed coordinates

r ; , r ; . We c o ~ ~ l d

transform it to ordinary coordinates with the help of the brackets in (2.21).

d)

THE HARTREE-FOCK (HF) APPROXIMATION.

-

The hamiltonian (1.3) for a two particle system has the form

As the ground state is a singlet, and thus anti-symme- tric in the spins, we can propose for the configuration space part of the HF solution of (2.29) two particles in the same orbital i. e.

The standard analysis [13] gives, for the single particle

state $(r,), the equation

T H E PSEUDO-ATOM : A SOLUBLE M A N Y BOD\- PROBLEM C4- 129

This equation appears t o be integrodifferential, but

-

E' o - - +[(1

- &)I;'

+

⁽¹

+

& ) ' I 2 ] ,

( 2 . 4 1 ) actually, as $(r2) is of definite parity, the expectation

_-.

value of the term r , .I-, vanishes and thus it becomes

Lf/; =

T C - ~ / ~ ( L

- E )- ^{7 ' 8}

exp[- +(I -

E)"'

r;*]

x

+

^{(Z - 1) i-f]}

$ ( r l )

= e'

$ ( r l ) (2.32) x exp[-

3(1

+ ^&)'I2 . ( 2 . 4 2 )

where To carry out the perturbation approach with H F

(2.33) states, we must develop Eh and !Pb in a power series in

E

similar t o tlie development (2.14) in 6. For the energy tliis is trivial and we get for the coefficient Clearly then the H F average potential is also of the

of

&,,

we designate by Ep, values harmonic oscillator type but of frequency (Z - 1)'

rather than the Z' of the one body part of the hamil-

- -

3 ( 2 p - 3 ) ! ! E - 3 E

= - -

tonian (2.29). From (2.33) the energy levels in this

0 - Y p

2" p

! ,

potential are given by

-

- i ( Z - I)-'

(2 ¹²

+

1

+ 311 It is interesting to note that in this case only even

=

( Z -

1)% ( Z

-

-:-)

( Z

-

I)-' ( 2

¹¹

+ ^I + 3) , orders of the perturbation contribute to the energy.

( 2 . 3 4 ) For the eigenstate (2.42) we need first expand and the corresponding eigenstates become

$ooo((l k

&)%,

r )

³

(2.44) (2.35) in terms of $,,oo(r). This can be done as in (2.16) The HF energy of tlie ground state is given by the and combining terms we can expand Pb in a power expectation value of H of (2.29) with respect to the series where the coefficient of

E,,

which we designate state (2.30), in which $ ( r l ) is the lowest eigenstate by &,,, take the form

of (2.32). As the expectation value of r , .r2 is again zero, we obtain

(2.36) where Once we have the HF common potential for tlie two 1

particle system, the corresponding complete family

^{M;',I;? =}

1

^(- ~ ) 1 " + ' ' 2 r l l Y 2

of eigenstates is given by

We can then use these states to carry out a pertur-

( 2 n ; + ^{2 9 , -} I ) ! (212; + ^{2 9 , - I ) !}

bational or variational analysis for the exact problem.

-

Before proceeding with this program we carry out q 1

!

q2 ! ( 2 n; + ^{q , )}

^!

⁽²

^n;

+ ^q,)

^!

a scale transformation

with q,, 9 , restricted by ri

⁴

(2

- 1 ) - ' I 4

ri ,

i =

1, 2 , (2.38)

q1 + ^q2 +

^11;

+ ^n;

⁼ p ,

q1 ,<

p, 92

< ^P .

on tlie HF wave functions (2.37), so that they become

states of frequency

w =

1 . The same transformation As in

b,

we now define

applied to the hamiltonian (2.29) will give us Ep = Eo +

^-

^{E , E} + ... +

^-

E ~ E ~ ,

=

+

^{(1 -} E ) I ' ; ~ ]

+ ^+[pi2 +

⁽¹

+

^{E )}

^ri2]

^,

( 2 . 3 9 ) and plot as function of p both the energy ratios and

where

E =

( Z

- ] ) - I .

(2.40) We shall use tlie liamiltonian in the following sections, as in tliis way we avoid botliersome factors in any expansion of the exact solution in terms of HF states.

e)

THE

PERTURBATION APPROACH ^WITH

H F STATES.

- The ground state energy and wave function of the hamiltonian of (2.39) are given by

overlaps with the exact solution that were defined in (2.23). Numerical results for up to

p =

10 will be given in section

i).

Again we note that our perturbational wave func- tions are expressed in terms of the

0,

of ( 2 . 4 9 , which depend on the coordinates r ; ,

r;.

If we want toexpress them in terms of states depending on r,,

r,

we must use the transformation brackets in (2.21).

f ) THE

VARIATIONAL APPROACH WITH H F STATES. -

As in section

c)

the variational approach require the

C4-130

M. MOSHINSKY, 0 . NOVARO A N D A. CALLES

calcglation of the matrix of with respect to the lent to considering the set of states (2.21) in which states (2.25). If we limit the matrix to agiven maximum

2(n; + ^{n;) 6}

^p

. ^(2.48)

number of quanta that we denote by

p,

it is equiva-

We thus need only to calculate the matrix elements (ii; 0 n; 0 00 1 XI n; 0 n; 0 00)

=

(2

^11;

+ 2 ni + ^{3) d;;,; -}

Diagonalizing the matrix in which (2.48) holds for both n;, n; and

ii;, n;

gives us, among others, the ground state energy and normalized wave function, the latter in terms of HF states. Numerical results will be discussed in section

i ) .

g) THE MINIMIZATION OF THE

VARIANCE. - SO far we have discussed only variational approaches based on the expectation value of the hamiltonian using either non-interacting or HF complete sets of two particle states. Another possibility is the variational approach based on the variance [6, 71 i. e. the expec- tation value of the operator

where H is the hamiltonian of our problem (in the present case (2.29)) and E the ground state energy (in the present case zli Eh or

( 2

- 1) 1/2& depending on whether we use non-interacting or HF states).

As H is hermitian, the operator (2.50) has clearly only non-negative eigenvalues, and if we are able to find the exact eigenstates one of them would be the ground state wave function of our problem corres- ponding to the eigenvalue zero of (H - ^{E ) ~ .}

We could now carry out a variational analysis for ( H - E)2 starting either from the non-interacting or HF states for u p to

p

quanta. The matrix elements of (H - I?)' _with respect to these states can be imme- diately obtained either from (2.28) or (2.49) and thus the eigenvalues and eigenstates of the operator (2.50) can be derived. The lowest eigenvalue gives the minimum of the variance and it clearly should approach zero when

p -,

co. The ground state energy associated with the minimum variance can be obtained by considering the expectation value of the hamiltonian with respect to the corresponding eigenstate of (H - E)'. We can compare the variance results for ground state energy and wave function, which are a function of p, with the exact solution. This will be done in section

i)

for up to

p =

10 quanta.

In an actual physical problem the variance analysis is implemented by substituting for E in (2.50), the experimental value of the ground state energy.

17)

UPPER

A N D

LOWER BOUNDS

FOR THE

GROUND STATE ENERGY. - Variational analysis provide upper

bounds for the ground state energy. It is very conve- nient to be able to estimate lower bounds also, as then we can bracket the exact energy in a narrowing region as we increase the degree of approximation.

Many lower bound criteria exist. Here we sl~all only mention a criterium developed by Temple [I41 as modified by Conroy [7], which requires the know- ledge of the energy G , and the variance for the corres- ponding wave function

Y,

of any type of variational solution that gives upper bounds, plus the experimen- tal energy E, of the first excited state accessible to the same variational calculation. The lower bound is then given by the formula

where U: is the variance, i. e. the expectation value of the operator (2.50) with respect to

Y',,

and we desi- gnate by E, the exact energy of the ground state.

If we take Y, to be the wave function corresponding to the minimum value of the variance, then the value of U: for a variational analysis going up to

p

quanta was obtained in the previous section and so we could immediately determine the lower bound. Numerical results will be discussed in the next section.

i)

NUMERICAL RESULTS

^FOR THE

DIFFERENT APPRO-

XIMATION

METHODS. - We proceed now t o discuss numerical results for PLi' when we apply the different approximation methods we proposed.

Let us first consider the perturbation procedures both with the non interacting and HF states. In figure 2 we plot the energy deviation from the exact value (defined in (2.230)) as a function of the order of the perturbation p. The deviation is given i n a logarithmic scale that goes from 1 to lo-". We see that the perturbation series converges rapidly but much more so for HF than for non-interacting states. The same holds for the deviation from 1 of the overlap of the exact and approximate wave functions given by (2.236), as seen in figure

3. In fact, the same order of appro-

ximation seems to be achieved in 4th order perturba- tion theory with HF states as in 8th order for non- interacting states.

In figure 4 we consider the energy deviations as

Function of the number of quanta p for the variational

T H E PSEUDO-ATOM : A SOLUBLE MANY BODY PROBLEM C4-131

.

I PERTURBATIONS HF I

PPERTURBATIONS STANDARD

ORDER OF PERTURBATION P

IVARIATIONAL HARTREROCK 2VARlATlDNAL STANDARD 3.VARIANCE H F 4.VARIANCE STANDARD

-

NUMBER OF QUANTA P

" " 0 4 6 8 9 ; o

FIG.

2. - Deviation ratio of the pth order perturbation energy

FIG.

4. - Deviation ratio of the variational and variance from the exact value for PLi+. The deviation in this and the energies for up to p quanta from the exact value for PLif.

following three figures are given in a logarithmic scale for both

HF

and non-interacting states.

I

I

^{ORDER OF} PERTURBATION t'

I

FIG.

3. -Deviation from 1 of the overlap between the exact and Pfl1 order perturbation wave functions for PLif.

and variance procedures both witli non-interacting and H F states and in figure 5 we d o the same for the deviation of the overlap from 1. Again tlie use of H F states improves matters considerably as, for example, we get the same degree of approximation when we go up to 6 quanta in the variational approximatioli witli H F states, as when we go u p to 10 quanta witli ordinary states. Results for tlie variance analysis differ very little from those of the variational one when we use the same states and number of quanta, though the variational is slightly better.

We complement figure 4 witli Tables I and

11.

I n Table I we give, for non interacting states, the upper bounds derived from variational a n d variance calcu- lations, as well as the lower bound of (2.51) as func- tion o f the number of quanta. We also give the variance minimum, i. e. tlie lowest eigenvalue of (H - E)' as a function of

p.

In Table I1 we d o tlie same thing for H F states. Note that the limiting values

NUMBER OF QUANTA P

0 1 2 3 4 5 6 7 8 9 1 0

FIG.

^5. - Deviation from I of the overlap between the exact and the variational and variance wave functions of PLi+ for

up to p quanta.

of the energy, when

p

increases, d o not agree in Tables I and

11.

This is due t o the fact that the exact values are EA and EA which are related by

Z S

EA

⁼

( Z

- I ) %

EA with Z

= 3

for PL;

Upper (uariational and variance) and lower boutfds of' the energy and the lninirnum value of the variance for PL~' as function of the number of quanta.

The non-inferacfion states were used

Number of quanta

- 0 2 4 6 8 10

Energy (varia- tional) - 2.500 001 2.384 941 2.368 395 2.366 285 2.366 052 2.366 029

Variance - 0.184 615 0.034 194 0.006 330 0.000 973 0.000 126 0.000 01 5

Energy (from variance)

- 2.500 001 2.388 617 2.369 076 2.366 351 2.366 057 2.366 029

Lower bound - 2.319 134 2.354 825 2.362 908 2 365 442 2.365 943 2.366 01 6

M . bIOSI4INSKY. 0. NOVAKO AND A. CAI-LES

Uppo (cariational arzcl cariatzce) atit/

101t.o

bo~1tid.v of' tllc energj. an(/ the tiiiiiitiilitii rallte o j ' the cariailce /or

PI,;'

as firtictiotz

of'

tllv rlltiiiber of' qt~atzta.

Tlle HF slates ,+.ere used Energy

(varia-

tional) Variance

- - -

0 3.000000 0.197 942 2 2.905 091 0.021 023 4 2.898 177 0.001 734 6 2.897 796 0.000 107 8 2.897 778 0.000 005 10 2.897 778 0.000 000

Energy (from variance)

- 3.000 000 2.906 400 2.898 245 2.897 798 2.897 778 2.897 778

Lower bound

-

2.849 128 2.890 148 2.896 95 1 2.897 720 2.837 774 2.897 778

Again we get a quicker narrowing of the gap between upper and lower bounds, and a faster conver- gence 01' the variance minimum to zero. when we use H F rather than non-interacting states. The convergence of the lower bound is quite satisfactory though it reaches the exact energy less pronlptly than the Ritz upper hound.

i ) DFKSITY M A T R I C I : ~

A N D

NATURAL ORRITALS.

-

As is well known [S], all the physical information on 11-particle systems that involve one o r at most two body operators can be obtained with the help of one and two particle density matrices. We shall derive these matrices for the two electron pseudo-atom and, in particular, discuss the correlation effects in the one body density matrix. We shall also obtain the matrix clements of the latter with respect to a complete set of one particle states. T h e diagonalization of the

-

matrix leads to the concept of natural orbital

[9].

This concept gives an insight into the most effective way 01' selecting the single particle wave functions that appear in the approximate determinantal states for our many body problem. Before proceeding with o u r problem we carry out the scale transformation

on the hamiltonian

( 2 . 2 9 )

to obtain

net

⁼

z- '

1-1 = $[p;

+ (z/Z)

^{r : ]}

+

+

^{f [ p :}

+ ( z @ )

^{r : ] - (2}

z)-'

^{( r l - r 2 ) ?}

= $[p',l

+

( 4 r ) , r i 2 ] i-

+

^{( 4 r i 2 ]}

,

( 2 . 5 3 ~ )

where

cr _= :[(Z - 2 ) / Z ] ' 1 2 ,

/I = p

(z;Z)"'. (2.53b)

This scale transformation will allow us t o discuss in a similar fashion the expansion of the density matrix in terms of non-interacting states (2

⁼

Z), as in terms of the HF states (Z

=

Z - 1).

T h e exact two particle solution can now be written as

and the two particle density matrix has the form

where the primed coordinates in the wave functions are expressed in terms of the ordinary ones through the relations ( 2 . 10).

T h e more interesting one body density matrix is obtained t l ~ r o u g h the definition

which, carrying out the elementary integrations, has the explicit form

where, for later notatio~zal convenience, we liavc replaced

i.,

by

r,.

The last factor in

( 2 . 5 7 )

shows clearly the correlation effect d u e to the interaction, as it cannot be written as a product of independent functions of

r 1

a n d

r,.

T h e density matrix

( 2 . 5 7 )

has two c o n t i n u o u ~ indices

r ,

and

r,.

It is of interest to express it in terms of a discrete set of indices, and we can use for this purpose the complete set of single particle states

cC/,,r,,,(r).

We define then the matrix

As

y ( r , ) r , )

of

(2.57) is

invariant under rotations, we see that we can write

. I

~ , l ~ / ~ n l ~ , n , / l n r l = I'nlnl 6 1 ~ 1 ~ ~ , n 2 t n I

,

( 2 . 5 9 )

where

in which the square bracket with index 0 indicates the coupling of the angular momenta o f the states of

r l

and

r 2

t o zero.

Using now the relative and center of mass coordi- nates,

in the expression (2.57) for

y(r,

1 r,), a n d taking

advantage of the transformation brackets introduced

in (2.21) and the value of the integrals given in

( 2 . 1 7 ) ,

we obtain for Y,',,,, o f

( 2 . 6 0 )

the following expression

From the properties [l I] of the transformation brackets, y!,,,, is symmetric under exchange of

¹¹²

and n , and tlii~s (as tlie pllase convention of the bra- ckets gives them in real form) 11

y,,,,,, I

;I is a real sym- metric matrix. T h e density matrix(2.59) is then broken into real symmetric submatriccs along a diagonal associated with the angular niomentum I. Tllcse submatrices I! $,,,,, 1; are infinite dimensional, but if we limit our analysis to a m a x i r n ~ ~ m number of q u a n t a p

=

2 Ni.e. n , ,

11,

<

h',

we I~avefjnitematrices.

If we diagonalize them wc have the natural orbi- tals [9] as linear combinations of $,,,,,(r), cither for non-interacting o r H F states.

Each natural orbital is associated with a definite eigenvalue

ol'

the density submatrix I

y,,,,,, I

/ . If our two particle state llas n o corrclations, for example if it is given by a n independent particle wave function such as H F , the eigenvalucs of the density matrix [8]

are just 1 o r 0. T o the eigenvalue I correspond then the occupied and to 0 the empty states in the inde- pendent particle model. When there is correlation, the eigenvalucs of the density subrnatrix differ from 1

o r 0 and the extent of the difference is a good measure of the correlation.

If we consider thc full infinite submatrix 11 i~lfr,l, ¹¹

for definite

!,

its cigen\-alucs will be clearly independent of thc basis. It' we restrict tllough

^{) I , ,}

n2 < N some bases will give

a

better approximation to the natural orbital states than others. For example, if wc use tlie non-interacting basis (i. e. z

⁼ Z )

for PLi+, calcula- tions u p to

p = 10

gi\,e the eigenvalues of 1)

j)nLIII 0

I1 that are presented in Tnble IIIa, while in 'Table lIIb we indicate the coefficients in the expansion of the first natural orbital i n terms of the wave functions

~ / / , , ~ ~ ( r ) In Tables 1Vu and 1Vb we carry out the same :~nalysis starting from the H F basis (i. e.

Z

= % -

I). We sce that in the HF basis the eigen- values

^of'

li

y:,,,!

arc closer to 1 o r 0 than is the case I'or the non-interacting basis. Also for the first natural orbital, thc coeficient of $,,,(r) comes closer to 1 for the H F than for the non-interacting case. These results again indicate that the H F basis comes closer to the optimum single particle wave functions than other basis.

Eigenvulzres of the densir,, rnatri,u,for P L i f us ujiritction o f t h e iliaximum number of qlrai~ta in tile IIO~I-interoctiiig states ill tern~s of whiclz it \vas expanded Number

of quanta -

0 2 4 6 8 10

Coeficients of the non-interactii~y states in the expansion of the Jirst natural orbital of P L i f as a functiorl of'the maxintunz number of quanta

in

the approximation Number

of quanta -

0 2 4 6 8 10

First Second Third Fourth Fifth

-

- - -

0.918 891

0.944 552 0.000 299

0.945 149 0.000 326 0.000 00

0.945 162 0.000 328 0.000 00 0.000 00

0.945 163 0.000 328 0.000 00 0,000 00 I 0.000 00 0.945 163 0.000 328 0.000 001 0.000 001 0.000 00

Sixth

-

M. MOSHINSKY, 0. NOVARO AND A. CALLES

Number of quanta

- 0 2 4 6 8 10

Number of quanta

- 0 2 4 6 8 10

Eigenvalues of the density matrix for PL~' as a function of tlte maximum number of quanta in the HF states in terilzs of which it was expanded

First Second Third Fourth Fifth Sixth

-- ^A - - - -

0.943 332

0.945 160 0.000 326

0.945 163 0.000 328 0.000 00

0.945 163 0.000 328 0.000 00 0.000 00

0.945 163 0.000 328 0.000 00 0.000 00 0.000 00

0.945 163 0.000 328 0.000 00 0.000 00 0.000 00 0.000 00

Coeficients of the HF states in the expansion of the first natural orbital of PLi+ as a function of the maxinzunz number of quanta in the approximation

k ) THE HYLLERAS

APPROACH.

-

In the original work of Hylleras [I51 on the helium atom, a very efective variational procedure was introduced in which the trial wave function was a product of the ground state two particle hydrogenic function of an.

appropriate charge by a polynomial in the magnitude of the relative coordinate I r , - r2 I. It is interesting to note that in the case of the pseudo-atom this pro- cedure is fully justified by the exact solution (2.54), which can be written as

Y;(Y;, r;)

= n -

3'2(4 (4 81318

electron to the n-electron pseudo-atom with the restriction, discussed in the introduction, that

11

<

2.

As in the two particle case, we shall limit our analysis to the ground state of tlie pseudo-atom, and further- more consider only values of

11

in which subshells in the common harmonic oscillator potential are filled. In these cases the ground states will be non-degenerate with total orbital angular momentum L

⁼

0 and spin

S =

0. Besides, in the H F approximation for these states [13, 161 we can consider that tlie single particle wave functions will be those associated with spherically symmetric potentials. As discussed in reference [16], the subsliells are filled at numbers

= ( a / ~ ) ~ ' " 1 --(a-pj r;' +2(a-/?j2

⁷ⁱ⁴⁺ ...

}

^{11 =}

2 8 10 20 26 40 42 52 70 76 90 112 etc.

x 7r-312(4 /3)3'4 exp[-2 ~ ( r : +I-:)] . (2.63) 1 s 1 p 2 s I d 2

13

I f 3 s 2 d 1 g 3 p 2 f 1 11, The last factor in (2.63) is just the product of two

zero quanta harmonic oscillator states of frequency 4 /I associated with the coordinates

r.,,

r,, while tlie term in the curly bracket is an infinite series in powers of

which for

(a -

fi) 4 1 can be approximated by a polynomial. Thus Hylleras approach would be the natural one to follow in the discussion of the ground state of the two electron pseudo-atoms.

3. The n-electron pseudo-atom for the case of closed shells. -

We will now pass from the two

(3.1) where underneath each value of

i t

we indicate the level in the common harmonic oscillator potential that is filled for tlie corresponding number of electrons.

In this chapter we plan to carry out the following

analysis. First in section a ) we give the exact solution

for our problem taking the result from reference [16],

but presenting it in a notation that does not use

creation operators. We then discuss in section b)

the rigorous Hartree-Fock self consistent analysis

(HFS) for the case n

⁼

8, i. e. when the 1s

-

l p

shells are filled, and compare the results with those

obtained when the single particle states are of the

harmonic oscillator type. The frequency of tlie latter

will be taken from the minimization of the expectation

THE PSEUDO-ATOM : A SOLUBLE M A N Y BODY PROBLEkl C1- 135

value of the hamiltonian of the pseudo-atom and we denote the Hartree-Fock analysis using these harmonic oscillator single particle states as the Hartree-Fock approximation (HFA) for tliis problem. For

i r =

8 the H F S and H F A turn out to give very similar results.

In reference

[16]

we compared the energy and wave functions of the exact and H F A for the 11-electron single ionized pseudo-atoms for up t o 112 particles.

From the analysis of the

11 =

8 case we then conclude that it is likely tliat a closely similar relation holds between the exact and H F S energies and wave func- tions as the one that holds between the exact and HFA.

The comparison between the exact solution and the approximation with antisymmetrized non-interacting states is much poorer than between the exact and HFA, as already seen in the two electron pseudo- atom, and thus will not be discussed further here.

As in the two electron pseudo-atom, we could start a perturbation or variation approximation using the complete set of H F A states. We sliall see in section

c)

though that tliere are some problems in the construc- tion of this complete set for

11 > 4 tliat are related to

the antisynimetrization of the wave function. Thus tlie analysis of chapter 2 can only be extended straiglit- forwardly to the cases when

11 =

3 or 4, in which case we d o not expect the results to be essentially different from those of

11 =

2, and so we d o not carry out a numerical comparison for these cases.

What is much more interesting for the 11-particle problem is the discussion of correlation effects involv- ing more than two particles. We sliall outline in sec- tion

d)

a possible procedure for studying these effects in the case of

11 =

4 electrons with spin S

=

2, tliat correspond to a closed

1s

-

l p

sliell. Tlie analysis will be qualitative, as we shall leave for a future publi- cation a quantitative discussion of correlation effects.

Finally, in section

e)

we discuss the form factor for the exact ground state and compare it witli the value one gets in the H F A . This gives us an idea of the influence of correlation effects due to tlie interaction on the cliarge distribution in the pseudo-atom.

0)

THE EXACT SOLUTION

FOR A

CLOSED SHELL PSEUDO-ATOM.

^-

Let us first consider an 11-electron liamiltonian where tliere is no interaction and tlie charge is 2, i. e. the hamiltonian has the form

For the closed subshells indicated in (3.1) we have that the ground state is given by the determinantal function

-

Yz(ri ci)

=

(11

!)-"

det \I !Pnjfjmj(zx, ri) SPjn, 11 ^,

where S,,,, is the spin state uitli

^{cri =} f

4 the spin variable, and ,uj

=

1 + the spin projection. Tlie quan-

tum numbers n j l j

r , i j /ij

take all the possible values in tlie filled sl~ells, for example, for

11 =

8 we have

If now we have the Iiamiltonian

(1

.3) for the pseudo- atom, the analysis carried in reference

1161 shows that

the exact solution for the ground state can be written as

YeX =

'I/z-,,(ri a,) [Z/(Z

-

I I ) ] ~ ~ '

x

1

1 "

x e x p ( - ? [ ~ "

-

( ~ - n ) " ] - ( x

11 i = l

r,) ) .

As for tlie energy of tliis state, we showed in refe- rence

[16]

that it lias zero quanta in the center of mass coordinate and that the number of quanta in the Jacobi coordinates (1.4) is the same as that in the function

!Pz-,,.

The latter number we designate as (Il(11) and as can be seen immediately from the energy level diagram for the l~armonic oscillator it takes the values

Thus from (1.6) the energy for these ground states is given by

Having the exact ground state and energy, we now pass to tlie discussion of the same in H F S starting with

11 =

8, as the

11 =

2 case was already analyzed in cliapter 2.

6 ) T H E HARTREE-FOCI( SELF CONSISTENT ANALYSIS (HFS)

FOR T H E

CLOSED 1s

-

Ip SHELL.

^- 111

tlie case of tlie singlet ground state of the two electron pseudo- atom, which corresponds to a closed 1 s shell, we obtained the H F integro differential equation and solved it trivially, sliowing that the K F potential is of the harmonic oscillator type, but of frequency (Z

-

I)".

The siniplificatiori in this case is due to the fact that for the closed 1s shell tliere is no excliange term, while the ordinary one simplifies drastically when the interaction potential is of the harmonic oscillator type. T h e latter property still holds for closed sliells beyond Is, but the excliange term no longer vanishes and this prevents us from solving the integro differen- tial equations in the simple way of section cl), chapter 2.

Thus for the closed 1s

-

I p shell and beyond, we

shall not try t o derive and solve the H F integrn

differential equation but rather follow tlie familiar

H F analysis employed in nuclear physics [13]. In this

analysis one proposes a determinantal function of the

C4-136 M. MOSHINSKY, 0. NOVARO A N D A. CALLES

type (3.3) in which tile, as Yet undetermined, single Once 2 is known, we can construct a determinantal particle state in some central potential (remember we wave function witli the single particle states (3.8, 3.9) are dealill!? wit11 closed shells only [131), are desi- and take the expectation value of the liamiltonian (1.3)

gnated by with respect to it. This expectation value will give rise

~ , b ~ , ~ ~ ( r ) J ~ ~ , ~ ~

=

0, I, 2, ... (3.8) to polynomials of second and fourth degree (the latter due to the two particle interactions) in tlie coeffi- We can develope these states in terms of harmonic

cients c:(l). When we minimize this expression, i, e.

oscillator wave functions as follows

:

derive with respect to c:'(l), we get a system of algebraic

$vitll(r)

=

c:(l) $?l~!II("> 'I

⁷

^(3.9) equations in the coefficients where terms in first and where the coefficients c':(I) and the frequency Z~ are

as yet undetermined. I n the present version of the H F analysis, the problem rests in the evaluation of Zs

and c':(l).

For the frequency 2% a simple procedure suggests itself. Let us take the determinantal wave func- tion (3.3) and calculate the expectation of the hamil- tonian (1.3) with respect to it. Clearly this expectation value will be a function of Z, n and 2 and if we minimize it with respect to the latter, we get a reaso- nable value for the frequency 2%. This was done in reference [16] where we showed that 2 becomes

third degree in tlie c':(l) appear. Solving these equa- tions selfconsistently provides us then with the HFS solution of our problem.

The procedure has been fully discussed in refe- rence [13]. Here we shall apply it t o the 1 s

-

1 p closed shell pseudo-atom. As in this case we have only one state each for I

=

0 and I

=

1 (as in contrast for example, to the 1 s - 1 p

-

2 s shell where we would have two I

= 0

states), the index v becomes redundant and we can suppress it c:(l). Designating now tlie coefficients by

cn(l) - ^{c: 1}

⁼

^{0, 1 (3.11)}

-

=

- n $n[R(n) + 3 n]-' .

^.

(3.10) the coupled system of equations takes the form witli

where e, and el are the eigenvalues for the energies in .Is and l p shell, while the matrix elements appearing in the equations have the values

with

son between the exact and H F A for energies and wave H(I' I" I)

=

[(2

1'

+ 1) (2 1" + ^I)]'

^x

_{functions in} closed shells for n up to 112.

[ 4 n ( 2 1 + I ) ] - ~ < 1 ' 1 " 0 0 1 1 0 > , ( 3 . 1 3 ~ ) where <

⁽

> is a Clebsch-Gordan coefficient.

Solving the system of equation (3.12) self consis- tently when we go u p to ten quanta, i. e. n

=

0, 1, 2, 3, 4, 5, we obtain the coefficients shown in Tables 5,6 for c;f and c;. As

c:,

c(: are so large compared to the others, it is quite clear that HFS solution gives almost the same result as H F approximation (HFA) in whicl~ we take (3.3) as our determinantal wave function with 2, given by (3.10). The analysis for HFA

was

done in reference [16] and in tlie next sec- tion

wc

briefly summarize the results of the compari-

C) COMPARISON BETWEEN

THE

HARTREE-FOCI\:

APPROXIMATION (HFA)

^AND THE

EXACT SOLUTION.

-

As indicated in the previous section, the Hartree- Fock self consistent analysis gives almost the same result as the HFA. For the latter the wave function is given by (3.3) with 2 determined by (3.10). In tlie case of closed shells the energy EHF in the HFA was discussed in reference [I61 and its explicit expression is EllF

=

( Z - lt)' [(3I(l1) + 3

^{111 X}

x ( 1 + ³ 11[%(11) + +

^111-'

^{( Z - it)-'} 1%. ^(3.14)

Coeflicic~tits c z i r [lie e.vparisiori of'the 1

s

HFstatc~ in P F f iri terttis of'liart~iotiic oscilla~ors states Number

of quanta

-

0 2 4 6 8 10

Coefficients c; ill tlie expatisiori of the 1 p HF state

itz

PF' in tertns of hartnotiic oscillator states offiequency Zs, asji~ticrioti

qf

the nzaxinium number of quanta in the approximation Number

of quanta

-

0 2 4 6 8 10

As the exact energy and wave functions are given in

^I

i

F X l O 0 (RKHT SCALE)

⁷

^I5

I

^{FIG. 7. -} Overlap between the exact and H F wave functions

(3.7) and (3.54, we can compare them with the corres-

ponding expressions for the HFA. This was done in reference [16] and we just reproduce in figure 6 and 7 the difference in energies divided by E and the overlap,

1.

e.

when Z -

^{tt =} 1 ,

i. e. for single ionized pseudo-atoms W e notice from figure 6 that through the range of Z from 3 to 11 3, the error in the energy ( 3 . 1 5 ~ ) remains of the order of 10 %. The error goes to zero when

+ OVERLAP

'

\

+

\ + \

50.

40.

I

^{+ +} p s r u o b r ~ E v E a r s _{! 1}

or at most two body operators, a better measure of

i

^{9 I 21. 2.7 4 i 3 53}

RI' P f m o ' PSc RR~PTC' P I + ~lG'P7fr. : i s

how good is the wave function is given by

+ _{+ t ..} ^t ' for single ionized closed shell pseudo-atoms of up to 112 particles.

/ +---- + .- .. .

+' ^{+ . t -. ..+}

,

+ ? I 0

+

Z

-+ c ~ ,

as can be seen from (3.14) and (3.7), but this does not happen in the range of physical interest. As

2 1/11

[I

^('ffcx7 '+'F)

1 ] , (3.16)

FIG. 6. - Difference and deviation ratio of the H F energy

from the value for single ionized closed sljell pseudo-atoms

which ~ ~ ~ ~ 1 1 l i ; l l l y Ine~ISLIrCS the ~ v e r l i ~ p per pilrticlc.

of up to 1 1 2 particles.

It is clear then that the overlap ( 3 . 1%) may be cliiitc

-

for the wave functions we notice from figure 7 that the

s

overlap diminishes quickly as Z increases being 94

for Z

=

3, but only 2

"/,

when Z

⁼

113. It is impor-

tant to notice though that if we are restricted to one,

C4-138 M . MOSHINSKY. 0. NOVARO AND A . CALLES

small, while (3.16) remains very close to 1, if n is large enough.

If we could construct in a systematic way all excited states

9 5

with 2 given by ( 3 . lo), we could start with their help a perturbation or variation approach as was done in sections e) and j') of chapter 2, for the two electron pseudo-atom. Unfortunately the systematic construction of all it-particle harmonic oscillator states with definite spin (i. e. corresponding to definite irreducible representations of the permutation group of the coordinates), has only been done for

11 =

3 and 4.

Thus an analysis similar to sections e),

f )

of chapter 2 can only be carried in those cases. We do no expect though, for n

⁼

3, 4, that the results be substantially different from those obtained in tlie case n

=

2, and thus tlie numerical analysis will not be carried out explicitly.

For the many electron problem there are more interesting questions to be answered than the conver- gence of perturbation or variational procedures, among which correlation effects involving more than two particles are particularly important. We shall address ourselves qualitatively to this question in tlie next section.

L / )

CORRELATION EFFECTS INVOLVING MORE

^THAN

TWO PARTICLES. THE STATE

^OF

SPIN S

⁼

2

FOR11 =

4 PARTICLES.

-

We mentioned in the previous section that harmonic oscillator states of arbitrary spin have been obtained in a systematic fashion for

11 =

4 particles [17]. We could then, at least in this case, study correlations of more than two particles by developing the exact ground state in terms of a complete set of states of tlie HF type that belong to the same spin value. As we like still to deal with closed shell pseudo-atoms, to avoid tlie effect on HF solu- tions of possible deformed single particle potentials, it is particularly interesting to discuss tlie state of lowest excitation for n

=

4 that has spin S

=

2. This state corresponds to a closed 1s

-

l p shell in which all the spins of the particles are parallel, i. e. in configu- ration space the state is characterized by the irredu- cible representation { l 4 ) of the symmetric group S(4).

The exact ground state for n

=

4, S

⁼

2, and the complete set of HF states belonging to the I R {

l 4

)

of S(4) have been given in reference [17]. The H F states could be divided into sets in which only one particle, only two, only three or all four particles are excited.

The expansion of the exact ground state in terms of HF states belonging to the different sets, give us coeffi- cients which provide a measure of how importai~t are one, two, three or all four particle correlations. We plan to discuss this problem in detail in another paper.

It is of interest also to consider for tlie n

⁼

4, S

=

2, problem the difference of density matrices

For the HF state this difference is zero, but for the exact wave function it gives a measure of the correla- tion which we also plan to discuss in the paper just mentioned.

e) FORM FACTOR

FOR T H E

it-ELECTRON PROBLEM.

-

An important problem in atomic theory is tlie deter- mination of the effect that the interaction between the electrons has

011

the charge distribution in the atom, or equivalently, on its form factor defined by

We proceed to show that for the ground state (3.5), the form factor F(q) is very easy to determine. We notice first that the exact solution (3.5) could also be written as

Ye/,,

-

^Qz

^-,,(rir

^{C T ~ ) - 3/4 z318}

^{exp(- t z'I2 rk2) ,} _{(3 .19)}

where

The function

@,-,(ri,

a i ) when expressed in terms of the Jacobi position vectors (1.4), turns out to be independent of the center of mass coordinate r,: 1161.

Substituting (3.19) in (3.18), we can write the form factor as

x

1 exp(in-'"

q . ri) n-)" z3I4

exp(- z"' r r ) dr,', , (3.21)

because

^u,

- n - % r,', depends only on the relative coordinates r;, ..., rk-1. Multiplying and dividing

(3.21) by