ARTICLES
Quantum Monte Carlo: Direct calculation of corrections to trial wave
functions and their energies
James B. Anderson
Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802
͑Received 4 January 2000; accepted 10 March 2000͒
We report an improved Monte Carlo method for calculating the difference
␦
between a true wave
function ⌿ and an analytic trial wave function ⌿
0
. The method also produces a correction to the
expectation value of the energy for the trial function. The nodes of the trial function are not
corrected and the energy is corrected to the fixed-node energy of the trial function. Applications to
several sample problems as well as to the water molecule are described. © 2000 American
Institute of Physics. ͓S0021-9606͑00͒30921-7͔
I. INTRODUCTION
We have previously described a quantum Monte Carlo
͑QMC͒ method for the direct calculation of corrections to
trial wave functions.
1–3
We report here an improved method
which is much simpler to use. Like its predecessors the im-
proved method gives ͑for fixed nodes͒ the difference
␦
be-
tween a true wave function ⌿ and a trial wave function ⌿
0
,
but it gives in addition the difference between the true en-
ergy E and the expectation value of the energy E
var
for the
trial wave function.
The statistical or sampling errors associated with the
Monte Carlo procedures as well as any systematic errors oc-
cur only in the corrections. Thus very accurate wave func-
tions and energies may be corrected with very simple calcu-
lations.
For systems with nodes, the nodes are unchanged. The
wave functions and energies for these systems are corrected
to the fixed-node values—those corresponding to the exact
solutions for the fixed nodes of the trial wave functions.
The method has the very desirable features of: good
wave function in/better wave function out ... good energy
in/better energy out.
II. THEORETICAL BASIS
The diffusion quantum Monte Carlo method is based on
the simulation of the time-independent Schro
¨
dinger equation
in imaginary time
4,5
according to
ץ
⌿
ץ
ϭ
ប
2
2m
ٌ
2
⌿ϪV⌿. ͑1͒
With incorporation of importance sampling
6
based on a trial
function ⌿
0
, the above equation becomes ͑in atomic units
with m equal to the mass of the electron͒
ץ
f
ץ
ϭ
1
2
ٌ
2
f Ϫٌ•
͑
f ٌ ln ⌿
0
͒
Ϫ
ͩ
H⌿
0
⌿
0
ϪE
ref
ͪ
f, ͑2͒
where f ϭ⌿⌿
0
, the product of the true wave function and
the trial wave function, and E
ref
is a reference energy.
Making use of the difference
␦
ϭ⌿Ϫ⌿
0
and defining
another function gϭ
␦
⌿
0
ϭ(⌿Ϫ⌿
0
)⌿
0
we obtain
2
ץ
g
ץ
ϭ
1
2
ٌ
2
gϪٌ•
͑
gٌ ln ⌿
0
͒
Ϫ
ͩ
H⌿
0
⌿
0
ϪE
ref
ͪ
g
ϩ
ͫ
Ϫ
ͩ
H⌿
0
⌿
0
ϪE
ref
ͪ
⌿
0
2
ͬ
. ͑3͒
The term H⌿
0
/⌿
0
is the local energy E
loc
for the trial wave
function. The last term in Eq. ͑3͒ is a distributed source term
S, which may be positive or negative. It is convenient to
introduce the expectation value of the energy E
var
for the
trial function and write S as a collection of terms
Sϭ
͓
Ϫ
͑
E
loc
ϪE
var
͒
⌿
0
2
͔
p
ϩ
͓
Ϫ
͑
E
loc
ϪE
var
͒
⌿
0
2
͔
n
ϩ
͓
Ϫ
͑
E
var
ϪE
ref
͒
⌿
0
2
͔
͑4͒
or
SϭS
p
ϩS
n
ϩS
q
, ͑5͒
where
S
p
ϭ
͓
Ϫ
͑
E
loc
ϪE
var
͒
⌿
0
2
͔
p
,
S
n
ϭ
͓
Ϫ
͑
E
loc
ϪE
var
͒
⌿
0
2
͔
n
,
S
q
ϭ
͓
Ϫ
͑
E
var
ϪE
ref
͒
⌿
0
2
͔
,
and where the subscript p indicates a region of E
loc
ϽE
var
and
a positive particle feed, the subscript n indicates a region of
E
loc
ϾE
var
and a negative particle feed, and the subscript q
indicates an additional particle feed, normally negative.
In the procedure used previously, Eq. ͑3͒ was simulated
by g-particles fed continuously to the system according to
the source terms; allowed to diffuse, drift, and multiply or
disappear; and to cancel each other, positive with negative,
regardless of position if beyond a specified age or elapsed
JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 22 8 JUNE 2000
96990021-9606/2000/112(22)/9699/4/$17.00 © 2000 American Institute of Physics