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### Submitted on 1 Jan 1978

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**A MONTE-CARLO METHOD OF CALCULATING** **THE ENERGY OF 3He**

### R. Clark

**To cite this version:**

### R. Clark. A MONTE-CARLO METHOD OF CALCULATING THE ENERGY OF 3He. Journal de

### Physique Colloques, 1978, 39 (C6), pp.C6-121-C6-122. �10.1051/jphyscol:1978655�. �jpa-00218038�

**JOURNAL ****DE ****PHYSIQUE ** **Colloque C6, suppl6ment au no 8, Tome 39, aoOt 1978, Page ****C6-121 **

### A

MONTE-CARL0### METHOD

OF CALCULATING### THE

ENERGY OF 3 ~ eR.C. Clark

**Department of Natural Philosophy, University of Aberdeen, Scotland **

Rdsum6.- La fonction d'onde de Jastrow permet 1 1'Cnergi.e d'un systsme de ferm?on d'8tre calculde
par les techniques de Monte-Carlo. Cet expos6 prdsente quelques rdsultats prdliminaires concernant
3 ~ e **1 la densi'td expdrimentale et ddmontre qu'une loi d'dchelle permet de ddduire des rdsultats 1 **
des densitds diffsrentes.

Abstract.- The Jastrow wave-function allows the energy of a fermion system to be calculated by Monte Carlo techniques. Some preliminary results are presented for 3 ~ e at the experimental density and it is shown that a scaling relation allows results at different densities to be deduced.

The use of a Jastrow product wave-function where $ is real and

### 4

is a Slater determi-0

nant of plane-wave orbitals and spin functions,al-
lows the energy of 3 ~ e to be expressed as a classi-
cal average and hence Monte-Carlo techniques may be
used/ **I / . **

The method given below differs from that al- ready published by Ceperley, Chester and Kalos/2/

in that a slightly simpler computational procedure is used and spin-flipping is allowed. The use of a

"Green's relation" results in a larger variance of the average energy so convergence is not as rapid but the method permits the use of a scaling rela- tion so that results at any density may be deduced from those at a standard density.

The energy of the system is <H>=/~T$+H$ /TI

### L2

^{F }

^{F }**where H= **

### - -

CV~+.F.V. with Vij a pair potential +2m n n 1**J**ij

and q=/drJIF$. The integral over r denotes an inte- gral over all spatial coordinates and a sum over spins. We can write

<H> =

### - B ^{z }

d~ {$;m+v;$ + ### m+q,qov;qo

2m n

### I

### +

24,0vn$o.4 +### vn+t

+ <**C V**>

i<j ij (1 Since v2$=-k2+, a11 terms in (1) are real and hence

**n ** n

## I

2 dr$ovn$o

### .$+vn$

= (2)and with a little manipulation we can express <H>

The brackets on the right-hand side can now inter- preted as classical averages

<----> =

### I

d### -

r d^{spin }

### 1

P(rlS1; ~ 2 s ~ ;### ...

^{r S }

^{-N N }

^{)-- }

where P(rlS1; ~ 2 S 2 ;

### . . .

^{rNSN) }

^{= }

^{$:}

^{@+@/q }

The sum over spins and the integrals over

~1

### ...

r may be approximated by an average over a -NMonte-Carlo sequence as suggested by Metropolis, et al. 141. Such a calculation is not technically dif- ficult. Indeed the problem has been reduced essen- tially to one of writing a sufficiently rapid compu- ter programme.

The assumption that $ is a product of

### air

0

functions, i$jfij where f.. = f(rij), is now made.

**1 ****J **

In the boson problem McMillan /3/ showed that this reduces the problem to that of calculating the pair distribution function. Here the same reduction can- not be made. As noted in / I / the three-particle distribution function has to be introduced. Instead we have to calculate the average of C(Vn lln~l~)~ but this is easy to evaluate directly for each configu- ration of the tlonte-Carlo sequence so that calcula- tion of the kinetic energy presents no problem. As a check we have used this method on 'He and repro- duced McMillan's results satisfactorily.

The advantage of (3) over the equivalent expression published in /2/ is that it is clear that a scaling relation exists as in the boson system/3'/'.

Since the wave-numbers k scale with p1l3, it follows
that if S is defined by **p'= ** s3p and $o is taken to
depend on b/r (as will be used be1ow)then the energy
at density p' and variational parameter b'where

b' = b/S, in the special case of the Lennard-Jones potential, is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978655 Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978655

We have assumed that the parti'cle states fill a Fermi sphere and that each momentum state is occu- pied by two particles with opposite spins. For a given spin configuration the product $I+@ becomes the product of the determinants of two matrices, D l and D, of order N/2, one for each spin, whose typical element is C exp(k.r..) where r.. i's the vector

." ^{-13 }^{-1J }

r.-r. linking two particles of the same spin. The
**-3 ** **-1 **

sum is over all momenta (ihsi'de the Fermi sphere) which obey periodi'c boundary conditi'ons on the wall of the container. Si'nce the set of

### k

values does not change in the course of the calculati'on, this sum can be rapidly evaluated by using the symmetri'es of the set.In order to allow spi'n flippibg, two particles
of opposite spin are moved randomly and new spl'ns
assigned by randomly exchanging them between the
determinants. Only one column in each determinant
is changed and only the ratios of the new determi'-
nants D l and D2 to the old ones are needed, **A **neat
and rapid way of obtaining these i's to reduce the
matrices Dl and D,, which are positive definite,

real and symmetric, to L L ~ form where L is lower triangular (Cholesky reduction); then to shuffle the column to be changed to the last column by row and column interchanges, performing the same trans- formations to L while continually returning L to lower triangular form by unitary transformations.

Then only the last row of L need be recalculated and the ratio of the determinants is the ratio of the new final element of the last row to the old one !5/.

The Fermi repulsion between particles of like- spin means the trial configurations with the spins of two particles interchanged are unlikely to be accep- ted. Of configurations accepted only roughly one in eight has resulted from spin inte:change. This means that the inclusion of spin-flipping is probably un- necessary and the neglect of this in reference /2/

leads to negligible error.

The minimum energy we have obtained for 3 ~ e is
shown in table I. The averages have been taken with
162 particles over ". 350 000 configurations. The
density.is the experimental density **p **= 0.0164 atoms/

**l 3 **

**l 3**

^{and f }

^{(r) }

^{= }exp {-(bu/r)

**5,**where b is the varia-

Table I

Using the scaling relation this result is equiva-

**0 **

lent to an energy of -1.26 K at **p = **0.0142 a t o m s / ~ ~
and b = 1.03 which corresponds closely to the re-
sults given in /2/. We intend to publish later a
more systematic set of results.

I would like to acknowledge some practical advice from Professor J.P. Valleau (Toronto) on the running of Monte-Carlo programs, the help of the University of Aberdeen Computing Centre, and a re- feree for drawing my attention to reference 121.

References

/I/ Buchan, G.D. and Clark, R.C., J. Phys.

(1977) 3069. The fact that the energy of a fer- mion system amy be written as a classical ave- rage was realized in connection with the elec- tron gas by Becker, M.S., Broyles, A.A. and Dunn, T., Physics

*175 *

(1968) 224. This should
have been acknowledged in reference /I/.
/2/ Ceperley, D. Chester, G.V. and Kalos, M.H., Phys. Rev. (1977) 3081

/3/ Metfillan, W.L., Phys. Rev. (1965) 442 /4/ Metropolis, N., Rosenbluth, A.W., Rosenbluth,

M.N., Teller, A.H., and Teller, E., J. Chem.

Phys.

*2 *

(1953) 1087
/5/ This produce was suggested to me by Dr C. Silk of Aberdeen University Computing Centre.

tional parameter and 0 is the standard de Boer-

**0 **

Michels parameter, 2.556 A