D:\291218499.doc
05/28/16
Page 1 of 8
The Full Monte Carlo Code
Abstract
The purpose of this work is to evaluate Gaussian integrals both analytically and by Biased Selection Monte
Carlo methods in order to illustrate and test the Monte-Carlo methods.
Introduction
The biased selection method began as a method to evaluate partition functions1. By using it to estimate and
minimize the variance in the local energy it can be used to solve the Schroedinger and Dirac equations. It
has been used extensively to estimate quantum mechanical expectation values. At its heart it is an
integration scheme which should seriously be considered for evaluating any integral of more than about 6
dimensions. The best description of the method in its current form is in the book Recent Advances in
Quantum Monte Carlo Methods2. In the following sections our implementation of information sampling
know as biased as random, the mixed extension that allows one to choose with respect to two centers and
the single_average method which allows for multistage selection are described and tested against analytical
evaluations of Gaussian integrals.
The integrals
The general form of the integrals tested is
g zr1 r1 d r d r
b
  ,  , N , m, n 
3
m
1
n
12
3
1
N
F
G
H
d i IJ
K.
N
N N
 
exp   ri 2     ri  r j
i 1
i 1 j 1
3
The analytical evaluation of this integral is discussed in analytical4.doc .htm.
The first integral is normalization Integral of the form
ze c
ez c h j
hj
  exp 2a r12  r22  r32  r42  r52  r62 d

exp 2ax dx
2
1
(1)
F I
 G J  29809.1
H2a K
18
18
Biased As Random
This is importance sampling3 which is nothing more than a change of variables so that in
I
z zz
zbg
z bg
2
0
d
1
1
d

0
b g
r 2 f r ,  ,  dr (2)
A new variable is given by
r
y
0

0
g x dx
(3)
g x dx
So that
1
R. L. Coldwell,"Monte Carlo Evaluation of the Partition Function for a Hard Disk System", Phys. Rev. A
7, 270 (1973).
2
“Atomic Calculations Using Variational Monte-Carlo” , S.A. Alexander and R.L. Coldwell, pp 39-64 in
"Recent Advances in Quantum Monte Carlo Methods" edited by W.A. Lester,World Scientific Publishing
Co, Singapore. (1997)
3
J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, Methune & Co. Ltd, London, John
Wiley & Sons Inc, New York. pp. 57 -59
D:\291218499.doc
05/28/16
bg . (4)
x dx
zgbg
gr
dy

dr

0
This then makes
x dx
cbg hzgbg
I  zz
d d z
dy
gc
r bg
yh
It is convenient to define
gc
r bg
yh
wbg
y 
(6)
4r bg
y zgbg
x dx
2
1
1
0
1
0
f r y , ,

0
(5).

2
0
  2
(7)
  2   1
So that all integrals are from 0 to 1 yielding
I

f r y ,   ,  
ebg bg dij (8)
wbg
y
zzz
1
0
1
1
0
0
d d dy
The Monte Carlo estimate of I is
I
f
1

w
N
N
fi
w
i 1
(9)
i
The error estimate is given by
1
 
N
2
FF
fI
G
G
K
HHw J
2
f
w
2
I (10)
J
K
The weighting function g(r) was one left over from an atomic calculation on neon.
Page 2 of 8
D:\291218499.doc
05/28/16
Figure 1 Guiding function used in this work with r2 exp(-r) for comparison purposes.
Page 3 of 8
D:\291218499.doc
05/28/16
Page 4 of 8
Figure 2 Integral of g(r) versus r
Two important items need to be mentioned here. The actual g(r) is the straight line connecting the
points representing it, which is exactly what is plotted in figure 1. The subroutines setup and bli enable this
to be accomplished. A set of points to best represent the function desired a guiding function is found by
minimizing the difference between the function and a linear interpolation to the function using 65 points.
These values of ri, g(ri) and the integrals from 0 to g(ri) are stored, setup.for. Since the actual guiding
function is the straight line connecting these points the integral of g(r) is exactly the trap rule up to each
point and exactly quadratic between points. A random y is picked and multiplied times the largest integral
of g(r). This value is then used in a quadratic equation to find the value of r(y) as indicated in figure 2. The
details are shown in testing\BiasAsR.for. In the six electron example of (1), the above was applied to each
electron, so that
c b gh(11).
g bg
x dx
z
And then (1) is evaluated as a single 18 dimensional integral so that
F r IJ
F r by gIJ
expG
expG
H
K  1  H K(12)

N
w
 wby g
6
wi  
k 1
g h rk y k

k
0
6
6
2
k
k 1
6
k 1
k
N
i 1
k
k 1
i
2
k i
i
D:\291218499.doc
05/28/16
Page 5 of 8
In the last step in (12), the rki refers to a r for each electron
Configurations
R=(-exact)/err Err-errfrom large/Err
R=(-exact)/errfrom large

1024
21794 +- 3462
-2.3
-0.328
-1.6
4096
23942 +- 1848
-3.2
-0.278
-2.3
16384
27320 +- 1152
-2.2
-0.100
-1.9
64536
28479 +- 617
-2.2
-0.036
262144
29662 +- 327
-0.4
-0.022
1048576
29802 +- 163
-0.0
-0.019
4194304
29703 +- 81
-1.3
-0.012
16777216
29771 +- 41
-0.9
-0.025
67108864
29800 +- 20
-0.4
0
With this many random numbers, the "randomness" of the set can be a problem for some
calculations4. The method used here, first suggested by MacLaren and Marsaglia 5, is to insert numbers into
a table with a multiplicative congruential method with one multiplier and seed and to extract them from the
table with a multiplicative congruential method with a different multiplier and seed. These are in Fortran
as testing\random.for and in C as the pair C\random.cpp, and C\random.h.
Optimizing the Guiding Function
A detailed account of optimizing the Guiding function for wave function minimization given by
Alexander et al6 can also be used for simple Monte Carlo evaluations. The variance as given by equation
(10) can is explicitly given by
F Ff I  L1 f OI (13)
G
Hw J
KM
NN  w P
QJ
H G
K
The probability w serves to take this from a sum to an integral, so this can also be written as
I
r dr
1 F f bg
  G
 z
f bg
r dr J(14)
z
N H wbg
r
K
1 1
 
N N
2
N
2
N
2
i 1
i 1
2
2
2
At first glance, it seems that something must be wrong owing to the fact that the first term in the variance
still contains the weighting function. This is not wrong, the standard deviation per configuration depends
on the method used to select the configurations. As shown in reference 4, the practical implementation of
 i which can be varied to
this is to imagine a wi that was used to select a set of configurations and a 
minimize the variance with respect to the set of points previously selected. Thus equation (13) becomes
I (15)
L1  f O
f
bg F
 w  M
G
P
NN w QJ
H
K
1 1
  
N N
2
N
i 1
2
N
2
i 1
which can be minimized with respect to parameters in
 i .
The electron-electron guiding function
The expectation value of 1/rij is dominated by the low probability of finding ri next to rj, so it does
not actually give the problems that might be expected from its singular nature. However, just as there is an
r2 in equation (6) that makes evaluation of singular nuclear electron potentials very straightforward, it is
possible to put an rij2 in the weights, if configurations are selected with respect to the anti-parallel electrons.
4
R. L. Coldwell, "Correlational Defects in the Standard IBM 360 Random Number Generator and the
Classical Ideal Gas Correlation Function", J. Computational Phys. 14, 223 (1974).
5
M. D. MacLaren and G. Marsaglia, J. Assoc. Comput. Mach. 12 (1965),83
6
S.A. Alexander, R.L. Coldwell and J.D. Morgan III, "Guiding Function Optimization in Biased-Selection
Monte Carlo Calculations" J. Chem. Phys. 97, 8407 (1992).
D:\291218499.doc
05/28/16
Page 6 of 8
In order to do this it is necessary to recognize that the weights in equation (6) are the probabilities relative
to uniform and random of having chosen the points in the manner that we did. As first proved in equation
28 of an early paper on the He-He potential7 by imagining a large lattice of T points and taking the
appropriate limits. If there are two ways of choosing a point x that proceed by probabilities w1(x) and by
w2(x), these two probabilities can be replaced by the average probability
x  w bg
x
(16)
bg w bg
2
wx 
1
2
This means that for any integral I 
f
, so that w is first averaged over all the ways of selecting the
w
configuration and then the inverse of this average multiplied by f is averaged. If it is not possible to get to
x with one guiding function, but is with another, obviously the non-contributor puts a zero in equation (16).
In the case of the electron-electron guiding function for carbon, what is desired is a guiding function that
explicitly removes the 1/rij between anti-parallel electrons with very little other effect. This is described in
more detail in eegfun.doc .htm and the code is in xnefin.for. The spin up electrons have been selected
before the first call to xnefin. For testing purposes the probability of selecting the next spin down electron
with respect to a spin up electron, pele, has been set in a data statement to 0.5. On entry to the code a
random number is used to decide if the selection is to be with the nuclear electron guiding function or with
the electron-electron guiding function. Then the electron's location is picked with the appropriate guiding
function. The average weight is given by the probability of selecting the electron with respect to the
nuclear electron guiding function plus the probability of selecting with respect to any of the up electrons
N
pele up

 
w r  (1  pele) wne rne 
wee r  ri (17)

N up i 1
bg
bg
c h
The following were run with pele=0.9
Configurations
Value of integral
Residual
Seed1
Seed2
1600
25900 +- 9827
-0.4
248
131
256000
31214 +- 2845
+0.5
248
131
4096000
28957 +- 731
+1.2
248
131
Choosing from a "single average"
It is possible to select an important configuration from a subset of M configurations 8.
1. Generate M sets of up electron coordinates xiup and weights wiup values using the biased as random
method above.
c h
2. For each set of xup values calculate the secondary weight function  xi
up
3. Generate a random number R such that 0 < R <

M
i 1

 xiup
ch
wc
x h
up
i
7
R.L. Coldwell and R.E. Lowther,"Monte-Carlo Calculations of the Born-Oppenheimer Potential Between
Two Helium Atoms Using Hylleraas Type Electronic Wave Functions", Int. J. of Quantum Chem. S12, 329
(1978).
8
S. A. Alexander and R.L. Coldwell, "Calculating atomic properties using variational Monte Carlo", J.
Chem. Phys. 103, 2572 (1995)
D:\291218499.doc
05/28/16
Page 7 of 8
4 Select a single configuration from the original set of M by finding the one at which

j
i 1

 xiup
c hfirst
wc
x h
up
i
becomes larger than R. The probability of generating xjup is given by

 2 x up
j

w x up

j
di
1
M

d i
 c
x h
wc
x h
2
M
i 1
up
i
up
i
The Fortran code config.for organizes this. It begins by getting Nup electron coordinates from the biased as
random generator xnfind. These are saved in N temporary locations. The determinant is calculated by a
separate subroutine WUP. The value of phi2(k) for each determinant and the sum up to k for each
determinant is saved. A random number is generated and locate called to determine which of the
configurations is to actually be used. Then the electron set and its total weight are set for the spin up
electrons. The locations of the spin down electrons are found using xnefin which a probability set in a data
statement of finding each spin down electron with respect to the spin up electrons rather than with respect
to the nucleus. This is also repeated M times and the final w for this is multiplied times the w for the spin
up electrons to compute the entire w. In the following tests,
electrons and
F
G
H
6
IJ
K
F
G
H
4
IJ
K
  exp .5 ri 2 for the spin up
i 1
  exp .5 ri 2 for the spin down electrons. It is expected to gain a small amount due
i 5
to the fact that the spin down electrons are placed preferentially with those spin up electrons that contribute
most significantly to the integral. The functions phi used in the single averaging are calculated in deter.for.
In the first test I use pele=0, so that only the single averaging is being tested. The random seeds
for all of the following tests are 45247, 89139.
Configurations
M
Value
Residual
1048576
1
29539 +- 162
- 1.7
262144
4
29596 +- 117
- 1.8
65536
16
29893 +- 103
+0.8
16384
64
29846 +- 100
+0.4
4096
256
29954 +- 100
+1.4
1024
1024
29870 +- 102
+0.6
In the second test I use M=64 and 16384 configurations, but vary pele from 0.1 to 0.999
Pele
Value
Residual
0.1
29722 +- 102
- 0.9
0.3
29898 +- 112
+0.8
0.7
29651 +- 188
- 0.8
0.95
30249 +- 1021
+0.4
0.99
30986 +- 5730
+0.2
0.999
5096 +- 551
-44.9
The standard deviations increase rapidly beyond 0.7. This comes about because there are regions
of the 6 dimensional space in which all electrons are separated by more than 0.5 a B. The electron electron
selection does not sample this region of space. Thus this part of the integration region is sampled only by
those rare events where the two spin down electrons are both chosen with respect to the nuclear electron
guiding function and in addition are selected more than 0.5 aB from all of the spin up electrons. When one
of these events happens, the probability w of its happening is small relative to the value of the integrand
and thus 3/w for this point becomes large enough to compensate for the usually missed region. This large
value causes the square to increase even more rapidly and gives rise to the increasing standard deviations
from pele = 0.95 to 0.99. At pele = 0.999, there are regions that are simply not sampled. Since the bulk of
the regions that are sampled appear correct, a standard deviation is given that compounds the problem by
D:\291218499.doc
05/28/16
Page 8 of 8
seeming to be accurate. This has been called the epsilon connection where a region of space with
probability epsilon of being selected actually contributes a value of R times epsilon which is significant.
The rij tests.doc .htm