The Exact Boundary Condition to Solve the Schrodinger Equation of Many
  Electron System
Microsoft Word - arxiv_prasadtext.doc
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 1 of 21)       
The Exact Boundary Condition to Solve the Schrödinger 
Equation of Many Electron System 
Rajendra Prasad 
Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India 
E-mail: Theochem@gmail.com 
In an attempt to bypass the sign problem in quantum Monte Carlo simulation of 
electronic systems within the framework of fixed node approach, we derive the exclusion 
principle “Two electrons can’t be at the same external isopotential surface 
simultaneously” using the first postulate of quantum mechanics. We propose the exact 
Coulomb-Exchange nodal surface i.e. the exact boundary condition to solve the non-
relativistic Schrödinger equation for the non-degenerate ground state of atoms and 
molecules. This boundary condition was applied to compute the ground state energies of 
N, Ne, Li2, Be2, B2, C2, N2, O2, F2, and H2O systems using diffusion Monte Carlo 
method. The ground state energies thus obtained agree well with the exact estimate of 
non-relativistic energies. 
INTRODUCTION 
An ideal target of a quantum chemist/physicist is to solve the non-relativistic 
Schrödinger equation exactly as it describes much of the world of chemistry. If we can 
solve this equation at a realistic cost, we can make very precise predictions. At present, 
only the full-CI method is available for obtaining the exact wave function within a given 
basis set, but this method is too demanding computationally and therefore not affordable 
even for a small system.  
In recent years increasing attention has been drawn to the random walk approach 
called diffusion Monte Carlo (DMC) method1 2 3 4 for solving Schrödinger equation. The 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 2 of 21)       
attractiveness of DMC method lies in that it can treat many body problems exactly. The 
DMC method is a projection method based on the combination of the imaginary time 
Schrödinger equation, generalized stochastic diffusion process, and Monte Carlo 
integration. The solution, it yields has only statistical error, which can be properly 
estimated and in principle, made as small as desired. Since in the DMC method the wave 
function has to be a population density, therefore, the DMC method can only describe the 
constant sign solution of the Schrödinger equation. This poses a serious problem if one is 
interested in the solution of a many electron system where the wave function is known to 
be antisymmetric (i.e. both positive and negative) with respect to interchange of two 
electrons. This situation is known as fermion sign problem in the quantum Monte Carlo 
literature1-4. The solution of this problem is one of the most outstanding in all of the 
computational physics/chemistry. This problem is often (mis)understood as a technical 
detail that defeating the numerical simulators. To the best of our knowledge no 
methodology is available to handle this problem in a systematic and controlled fashion. 
However, we think that it is essentially a problem of exact boundary, which is not known 
for many electron systems for obtaining well-behaved solutions of non-relativistic 
Schrödinger equation. It is our understanding that the boundary must be derived from the 
link between the formal mathematics and the physics of the real world. 
In this article, we will derive the boundary condition for atomic and molecular 
systems to obtain well-behaved solutions (i.e. bound state solution is single valued, 
continuous, quadratically integrable, and differentiable) of non-relativistic electronic 
Schrödinger equation. To start with, we are dealing with situations in which the ground 
state is non-degenerate only.  
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 3 of 21)       
THE EXACT BOUNDARY CONDITION 
We have the time independent Schrödinger equation: 
ˆ Ψ=Ψ EH                                                                                     …………..(1) 
where Ĥ  is the time independent non-relativistic electronic Hamiltonian operator in the 
Born-Oppenheimer approximation, E0 is the eigenvalue of the full many-electron ground 
state 0Ψ . The Ĥ  is defined in atomic units as follows: 
+−∇−=
electrons
electrons
electrons
2 1)(
1ˆ r                                                 ………..(2) 
where the external potential, ∑=
Nuclei
rV )(
,                                                 ………….(3) 
2∇  is Laplacian, ZI denotes nuclear charge, and  rIi and rij symbolize the electron-nucleus 
and electron-electron distance, respectively. 
Following Hohenberg-Kohn theorem I, a proof only of existence5, the electron 
density )(0 r
ρ  in the ground state 0Ψ is a functional of )(rV
, i.e. 
)(0 r
ρ  = )]([0 rV
ρ .                                                                     ………………………. (4) 
Further, the full many electron ground state 0Ψ  is unique functional of )(0 r
ρ , i.e. 
0Ψ = )]([ 00 r
ρΨ .                                                                                ……………………(5) 
Evidently we can say that 0Ψ  is a functional of )(rV
 i.e.  
0Ψ = )]([0 rV
Ψ .                                                                                    …………………. (6) 
We have a choice to express the exact density: 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 4 of 21)       
)]([0 rV
ρ  = ∑
2 )]([
φ                                                                          ………………..(7) 
Where N denotes number of electrons. The functionals, { }NirVi .....1)],([ =
φ  are exact 
ortho-normal one electron functions of the function )(rV
, which give exact )]([0 rV
(Caution to reader!! At the moment, here is nothing to do with so-called s, p, d, f, ..etc. 
type orbitals . The functionals { }NirVi .....1)],([ =
φ  are entirely different from those 
orbitals obtained from Kohn-Sham6 or similar formalisms.) 
Now we can write the exact N electron ground state wave function as a functional of N 
exact one electron functionals { }NirVi .....1)],([ =
    0Ψ = )]]([)],......,([)],([[
10 NN rVrVrV
φφφΨ                                                   ……(8)  
or 0Ψ = )]]([)],......,([)],([[ 22110 NN rVrVrV
φφφΨ                                                 ……..(9)  
Since each one electron functional in { }NirVi .....1)],([ =
φ  is a function of external 
potential )(rV
, we can also write the exact N electron ground state wave function in 
functional form as follows: 
    0Ψ = )](),......,(),([ 210 NrVrVrV
Ψ                                                …………………...(10) 
Thus the exact N electron non-degenerate ground state wave function is a unique 
functional of external potential experienced by each electron i.e. functional of 
)(),......,(),( 21 NrVrVrV
So far, it is not clear: 
• Whether the wave function is symmetric or antisymmetric with respect to 
interchange of any two electrons. 
• What are the analytical forms of { }NirVi .....1)],([ =
• What is the analytical form of the exact wave function? 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 5 of 21)       
• How to get the exact wave function from the exact density.  
However, we get some idea about the topology of a well behaved ground state non-
degenerate wave function and distribution functions in a given external potential )(rV
In particular: “The probability of n-electrons (where n = 2..N) being found 
simultaneously on the isopotential surface of an external potential )(rV
 is same 
irrespective of positions of the electrons on the surface.” 
Now we proceed to decide the nature (symmetric or antisymmetric with respect to 
interchange of any two electrons) of a well behaved many electron wave function.  
Defining the local energy, EL: 
∑∑ ∑ ∑
electrons
electrons
electrons
Nuclei
11 10
                                     ………….(11) 
The terms IiI rZ and ijr1  in the equation (11) will blow up if 0→Iir and 0→ijr  unless 
so-called cusp conditions are obeyed by 0Ψ . The 0Ψ  is exact and obeys electron nucleus 
(e-N) and electron-electron (e-e) cusp conditions.  
The wave function for a system of N identical particles must be symmetric or 
antisymmetric with regard to interchange of any two of the identical particles, i and j. 
Since the N particles are all identical, we could not have the wave function symmetric 
with regard to some interchanges and antisymmetric with regard to other interchanges. 
Thus the wave function of N identical particles must be either symmetric or 
antisymmetric with regard to every possible interchange of any two particles.  
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 6 of 21)       
 Let us assume 0Ψ is symmetric with regard to interchange of electrons i and j.  
There is a cusp in 0Ψ at rij = 0. This implies that 0Ψ  is not differentiable at rij = 0. 
Therefore, 0Ψ , symmetric with respect to interchange of any two electrons is not a well-
behaved solution. To make 0Ψ  a well-behaved wave function, 0Ψ  must be zero when rij 
= 0 and also it must change sign with respect to the interchange of two electrons, i.e. if 
ji rr
=  then 0Ψ  = 0.  This condition is universal and independent of kind of external 
potential. However, we are interested in a well-behaved solution of a bound state in a 
given external potential )(rV
. From the previous argument, we know that the 
simultaneous probability of finding two electrons is same everywhere at the isopotential 
surface. Therefore, if )( irV
- )( jrV
 = 0 then 0Ψ  = 0.  
Extending to N electron system, we have 
If ( ) 0)()(
=−= Π
rVrVf
 then 0Ψ  = 0. 
We can also express f  as Vandermonde determinant: 
)()(....)()(
)()(....)()(
)()(....)()(
11....11
rVrVrVrV
rVrVrVrV
rVrVrVrV
rVrVf
=−= Π               …(12) 
Consequently we have exclusion principle in the following form: 
“Two electrons can’t be at the same external isopotential surface simultaneously.” 
We see that if we are interested in a well behaved solution of the time 
independent Schrödinger equation, the boundary condition (12) (i.e. antisymmetric wave 
function) is obtained naturally due to singularity in the e-e interaction potential, which 
respects Pauli’s exclusion principle.  If electrons i and j are of opposite spin then we say 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 7 of 21)       
that )( irV
- )( jrV
 = 0 represents Coulomb (nodal) surface. If electrons i and j are of same 
spin then )( irV
- )( jrV
 = 0 represents Coulomb-Exchange nodal surface. All together, 
the ( ) 0)()(
=−= Π
rVrVf
 represents the Coulomb-Exchange nodal surface of N 
electron system. Hereafter, we will call f as ExchangeCoulombf −  nodal surface. However, the 
solution obtained for the Hamiltonian (2) within the boundary ExchangeCoulombf − =0 does not 
tell us about the spin multiplicity of the N electron system. 
Further, we can rewrite the functional f in terms of Hermite polynomials, )]([ rVH k
)]([)]([....)]([)]([
)]([)]([....)]([)]([
)]([)]([....)]([)]([
)]([)]([....)]([)]([
1112111
2122212
1112111
0102010
NNNNNN
rVHrVHrVHrVH
rVHrVHrVHrVH
rVHrVHrVHrVH
rVHrVHrVHrVH
−−−−−
=              …..(13) 
In particular, if we multiply an optimizable one electron functional )]([ rVψ  to the 
equation (13) and we obtain an N electron wave function: 
)]([)]([)]......([)]([)]([ 21 rVfrVrVrVNormrV N
rrrrr
ψψψ=Ψ   ……………………………(14) 
The one-electron density functional corresponds to the wave function (14): 
′′=′ ∑
1111 )]([)]([)]([)]([)]();([
kkk rVHrVHArVrVrVrV
rrrrrr
ψψρ      ……………….(15) 
where Ak is normalization constant of )]([)]([ rVHrV k
The two-electron density functional in terms of one-electron density functional: 
)]();([)]();([)]();([)]();([
)](),();(),([
21122211
rVrVrVrVrVrVrVrV
rVrVrVrV
′′−′′=
rrrrrrrr
……(16) 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 8 of 21)       
Here it appears that we can obtain exact ground state energy by optimizing only a one-
electron functional )]([ rVψ  in the equation (14). 
A very interesting and new physics obtained from the equation (13) is that each 
row in the determinant represents different level (k) of Kamalpur breathing (anharmonic 
quantum breathing) of electron cloud in a given external potential )(rV
 and each level, k 
is occupied by one electron (the elementary particle). 
BYPASSING THE SIGN PROBLEM 
We can bypass the fermion sign problem in the electronic structure diffusion 
Monte Carlo (DMC) method using fixed node approach. Here one assumes a prior 
knowledge of the nodal surface i.e. 
0(R) = 0. Due to tiling property
7 of the exact ground 
state wave function, the Schrödinger equation is solved in the volume embraced by the 
nodal surface, where the wave function has a constant sign and in this way the fermion 
sign problem is bypassed. The exact knowledge of Coulomb Exchange nodal surface 
allows us for an exact stochastic solution of the Schrödinger equation. The restriction in 
the random walk RR ′→  during the electronic structure diffusion Monte Carlo 
simulation is as follows: 
reject
accept
RfRf ExchangeCoulombExchangeCoulomb
)()(                                      ………….(17) 
We have applied the boundary condition (17) for the ground state electronic 
structure diffusion Monte Carlo simulation of N, Ne, Li2, Be2, B2, C2, N2, O2, F2, and 
H2O systems.  
Monte Carlo calculations can be carried out using sets of random points picked 
from any arbitrary probability distribution. The choice of distribution obviously makes a 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 9 of 21)       
difference to the efficiency of the method. If the Monte Carlo calculations are carried out 
using uniform probability distributions, very poor estimates of high-dimensional integrals 
are obtained, which is not a useful method of approximation. These problems are handled 
by introducing the importance sampling approach8 9.  In this approach the sampling 
points are chosen from a trial distribution, which concentrates on points where the trial 
function, ΦT(R) is large.  
 In the present DMC calculations, we have chosen the trial function, ΦT in the 
form: 
ΦT = Φ.F ,                                ….(18) 
where Φ denotes the Hartree Fock (HF) or multi configuration self consistent field 
(MCSCF) wave function and F is a correlation function that depends on inter-particle 
distances. The HF and MCSCF wave functions were obtained using the GAMESS 
package10 and employed Dunning’s cc-VTZ atomic basis set 11. In order to satisfy the 
electron nucleus (e-N) cusp condition, all Gaussian type s basis functions were replaced 
with eight Slater-type s basis functions. The exponents of Slater-type s functions were 
taken from Koga et al. 12 and satisfy the e-N cusp condition. 
We have chosen the Schmidt, Moskowitz, Boys, and Handy (SMBH) correlation 
function FSMBH
13. For the SMBH correlation function, Eqn. (19), we have included terms 
up to 2nd order, where order, s is defined as s = l + m + n. 
( ) 
+−= ∑∑ ∑
µµµµµ
atoms
electrons
iAASMBH rrrrrcF exp     ………….(19) 
where 
                                                                                        ………….(20) 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 10 of 21)       
and r denotes inter-particle distance. Six non-redundant parameters out of the total ten 
were optimized keeping b = 1.0 as follows: 
1) First we obtained optimal parameters by minimizing the energy and variance at 
the variational Monte Carlo (VMC) level. 
2) Using this VMC optimal trial function, the trial function fixed node DMC 
calculation was carried out and walkers were collected after each 2000 steps.  
Further, correlation parameters were reoptimized to minimize the variance with 
~100,000 walkers. Here reference energy was set to the trial function fixed node 
DMC energy.  
These optimized trial functions were used for importance sampling in the DMC 
simulation and a random walk RR ′→  was accepted if 
0)()( >′−− RfRf ExchangeCoulombExchangeCoulomb .  
The DMC calculations were performed using the open source quantum Monte 
Carlo program, ZORI14. Around 10,000 walkers were used for the systems studied. The 
Umrigar et al.15 algorithm was used for DMC walks and Caffarel Assaraf et al.16 
algorithm for population control. We have allowed only one electron walk at a time. The 
DMC calculations were done at several time steps. We report only those energies 
extrapolated to zero time step. 
 We present the ground state DMC energies of N, Ne, Li2, Be2, B2, C2, N2, O2, F2, 
and H2O systems in Table I. The DMC energies obtained using our newly derived 
boundary ExchangeCoulombf −  = 0 are far better than the trial function fixed node DMC 
energies17 and compare well with the experimental counterpart. However, present 
simulations were noisy and unpleasant compared to conventional trial function fixed 
node DMC simulations.  It is worth noting that we have obtained DMC energy even 
lower than the exact value at smaller time steps for the atoms of relatively larger atomic 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 11 of 21)       
radius perhaps due to failure of the distributions to reach the steady state or equilibrium 
distributions in a finite number of steps. This problem can be handled in the Green’s 
function quantum Monte Carlo (GFQMC) method as it takes the advantage of the 
properties of Green’s functions in eliminating time-step entirely in treating the steady 
state equation. The GFQMC is well suited if boundaries are exactly known18. If the trial 
function boundary and the ExchangeCoulombf − = 0 does not coincide and also non-zero values 
of trial function are very much different from the exact solution, which could lead to 
large statistical fluctuations from poor sampling and possibly to an effective non-ergodic 
diffusion process due to the finite projection time in practical calculations. Therefore, we 
are looking for an alternative well behaved trial function whose boundary coincides with 
those of ExchangeCoulombf − .  
CONCLUSION 
This article presents a progress of the author's research in order to get exact 
solution of non-relativistic Schrödinger equation of many electron systems. A conclusion 
of this on going research is that we have derived the exclusion principle “Two electrons 
can’t be at the same external isopotential surface simultaneously” using the first postulate 
of quantum mechanics. We propose the exact Coulomb-Exchange nodal surface i.e. the 
exact boundary to solve the non-relativistic Schrödinger equation for non-degenerate 
ground state of atoms and molecules. Using this newly derived boundary condition, one 
can bypass the fermion sign problem in the electronic structure Quantum Monte Carlo 
simulation and hence the exact ground state energy as well as the exact electron density. 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 12 of 21)       
                                                                                                                                                 
REFERENCES 
1.  Anderson, J. B. A random-walk simulation of the Schrödinger equation: +3H . J. Chem. 
Phys.  63, 1499-1503 (1975) 
2.  Ceperley, D. M. & Mitas, L. in New Methods in Quantum Mechanics, I. Prigogine, S. 
A. Rice, Eds.,(John Wiley and Sons, New York, 1996), Vol. 93. 
3. Hammond, B. L., Lester, Jr, W. A. & Reynolds, P. J. Monte Carlo Methods in Ab 
Initio Quantum Chemistry; World Scientific: Singapore, 1994. 
4.  Ceperley, D. & Alder, B. Quantum Monte Carlo. Science 231, 555-560 (1986) 
5.  Hohenberg,  P. & Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 136, B864-
B871 (1964) 
6.  Kohn, W. & Sham, L. J. Self-Consistent Equations Including Exchange and 
Correlation Effects. Phys. Rev. 140, A1133-A1138 (1965) 
7.  Cepereley, D. M. Fermion nodes. J. Stat. Phys. 63, 1237-1267 (1991) 
8.  Metropolis, N. A., Rosenbluth, W., Rosenbluth, M. N., Teller, A. H. & Teller, E. 
Equation of State Calculations by Fast Computing Machines. J. Chem. Phys. 21, 1087-
1092 (1953)  
9.  Grimm, R. C. & Storer, R. G. Monte-Carlo solution of Schrödinger's equation.  J. 
Comput. Phys. 7, 134-156 (1971) 
10.  Schmidt, M. W. et al. General atomic and molecular electronic structure system. J 
Comput Chem 14, 1347-1363 (1993)  
11.  Dunning, Jr., T. H. Gaussian basis sets for use in correlated molecular calculations. I. 
The atoms boron through neon and hydrogen.  J. Chem. Phys. 90, 1007-1023 (1989) 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 13 of 21)       
                                                                                                                                                 
12.  Koga, T., Kanayama, K., Watanabe, S. & Thakkar, A. J. Analytical Hartree-Fock 
wave functions subject to cusp and asymptotic constraints: He to Xe, Li+ to Cs+, H- to I-. 
Int. J. Quantum Chem. 71, 491-497 (1999)  
13. Schmidt, K. E. & Moskowitz, J. W. Correlated Monte Carlo wave functions for the 
atoms He through Ne.  J. Chem. Phys. 93, 4172-4178 (1990) 
14. Aspuru-Guzik , A. et al. Zori 1.0: A parallel quantum Monte Carlo electronic 
structure package. J. Comp. Chem. 26, 856-862 (2005) 
15.  Umrigar, C. J., Nightingale, M.P. & Runge, K.J. A diffusion Monte Carlo algorithm 
with very small time-step errors. J. Chem. Phys. 99,2865-2890 (1993) 
16.  Assaraf, R., Caffarel, M. & Khelif, A. Diffusion Monte Carlo methods with a fixed 
number of walkers. Phys. Rev. E. 61, 4566-4575 (2000) 
17.  Filippi, C. & Umrigar, C. J. Multiconfiguration wave functions for quantum Monte 
Carlo calculations of first-row diatomic molecules. J. Chem. Phys. 105, 213-226 (1996) 
18.  Kalos, M. H., Monte Carlo Calculations of the Ground State of Three- and Four-
Body Nuclei.  Phys. Rev. 128, 1791-1795 (1962) 
ACKNOWLEDGEMENTS 
The QMC calculations were carried out at the Lawrence Berkeley National Laboratory, 
Berkeley. The author gratefully acknowledges Professor W. A. Lester for his support 
during the stay at Berkeley. The author is indebted to Professor P. Chandra of Banaras 
Hindu University, Varanasi for his interest and helpful discussion during the preparation 
of the manuscript. Professor S. K. Sengupta of Banaras Hindu University, Varanasi is 
acknowledged for careful reading of the manuscript. 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 14 of 21)       
                                                                                                                                                 
TABLE I. The total ground state energies obtained from fixed-node DMC calculation.  
Atom / 
Molecule 
Bond 
length 
CSF,D ETFN-DMC 
(Ref. 17) 
ECEN-DMC 
(Extrapolated to τ=0) E0 
-, 111 
-54.5753(3) 
-54.5841(5) 
-54.5902(11) 
-54.5892 
Ne  1,1 -128.9216(15) -128.938(1) -128.9375 
Li2 5.051 
-14.9911(1) 
-14.9938(1) 
-14.9955(5) 
-14.9954 
Be2 4.63 
5,16 
-29.3176(4) 
-29.3301(2) 
-29.3378(15) 
-29.33854(5) 
B2 3.005 
6,11 
-49.3778(8) 
-49.3979(6) 
-49.41655(45) 
-49.415(2) 
C2 2.3481 
4,16 
77, 314  
-75.8613(8) 
-75.8901(7) 
-75.9035(9) 
-75.9229(19) 
-75.923(5) 
2.068 
4,17 
-, 552 
-109.487(1) 
-109.505(1) 
-109.520(3) 
-109.5424(15) 
-109.5423 
O2 2.282 
-150.268(1) 
-150.277(1) 
-150.3274(15) 
-150.3268 
F2 2.68 
-199.478(2) 
-199.487(1) 
-199.5289(25) 
-199.5299 
-199.52891(4) 
H2O  
-, 300 
-76.4175(4) 
-76.429(1) 
-76.4376(11) 
-76.438(3) 
-76.4376 
Bond lengths and energies are in atomic units. In the third column, we list the number of 
configuration state functions (CSFs) and number of determinants (D) in the trial function 
(ΦT). ETFN-DMC denotes the DMC energy with ΦT =0. ECEN-DMC denotes the DMC energy 
with ExchangeCoulombf − = 0. E0 denotes the exact, non-relativistic, infinite nuclear mass 
energy. The numbers shown in bracket are error bar. 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 15 of 21)       
                                                                                                                                                 
Supplementary note for reviewers: 
1. The proposed theory is to deal only real interacting many electron systems. To 
start with only non-degenerate ground state of many electron systems are 
considered. Author is neither interested nor intended to deal any kind of non-
interacting systems such as free fermion, free electron gases, or free particles 
because author think that none of the real system belong to either of these classes. 
Author has chosen to construct the boundary condition from the link between the 
formal mathematics and the physics of many electron systems. 
2. Difference between spatial nodes and Coulomb-Exchange nodal surface: 
I hope that people can distinguish spatial nodes and Coulomb Exchange nodes 
and the physics behind the different kind of nodes. Whatever I have discussed in 
this paper is only about Coulomb-Exchange nodal surfaces. There is no analogy 
with a particle in a box node and Coulomb-Exchange nodal surfaces. For 
example: The function f(r1,r2)=(r1-1)(r2-1)(r1-r2)exp(-2 r1-2 r2) is antisymmetric 
with respect to interchange of two electrons. However, the node (r1-1)(r2-1) is 
symmetric with respect to interchange of two electrons and fixed and this node 
can be compared with nodes of particle in a box. The Coulomb-Exchange node 
(r1-r2) is antisymmetric with respect to interchange of two electrons and 
responsible for removal of singularity in e-e interaction potential. The Coulomb-
Exchange nodal surfaces only occur in a system of more than one electron 
systems. Author understands that the Coulomb-Exchange nodal surfaces are 
directly responsible for the existence of real many electron systems. 
3. A consequence of proposed solution of the sign problem is that the ground 
state of Helium atom in the non-relativistic limit has a nodal surface.  
However, it is understood that He ground state wave function is symmetric 
and has no such node. 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 16 of 21)       
                                                                                                                                                 
A consequence of proposed solution for the sign problem is that the ground state 
of the Helium atom in the non-relativistic limit has Coulomb-Exchange nodal 
surface, r1-r2=0. 
An understanding that He atom ground state wave function is symmetric and has 
no such node is an illusion only. This illusion arises due to a practice that the 
QMC people using phi(1)*phi(2)*Jastrow trial function, where phi(r) is 1s orbital. 
The trial function is symmetric with respect to exchange of two electrons. The 
trial function also satisfies electron nucleus cusp condition. We also expect that 
the final solution will satisfy e-e cusp condition. Since the trial function is 
symmetric, people got accurate energy and assumed that the final solution is also 
symmetric and it does not have any node also wave function is non-zero at the 
point of coincidence of two electrons. Where is Coulomb hole? However, it can 
be proven that a symmetric solution is not acceptable. Proofs are as follows:  
“Proof for the existence of Coulomb-Exchange node in He ground 
state exact wave function” 
A.) Let assume  
 ),( 21 rrsym
Ψ  is an exact symmetric wave function. 
 i.e. ),( 21 rrsym
Ψ = ),( 12 rrsym
Since ),( 21 rrsym
Ψ  is exact, it must satisfy the cusp condition at 21 rr
= . Clearly 
there is a cusp at 21 rr
Since there is a cusp at 21 rr
=  in ),( 21 rrsym
Ψ , the second derivative 
2 ),( xrrsym ∂Ψ∂
 is not defined at 21 rr
Therefore ),( 21 rrsym
Ψ  is not a well behaved solution and hence it is not an 
acceptable wave function. Only option left is antisymmetric solution. 
B.) Another proof: 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 17 of 21)       
                                                                                                                                                 
 ),( 21 rr
Ψ  = ∑∑
νµµν φφ )()( 21 rrc
                                                  ………...(S-1) 
 Where { })(rrµφ  forms a real infinite one-electron basis. 
 Since ),( 21 rr
Ψ  is expanded over infinite basis set and hence it is exact. 
 This implies that ),( 21 rr
Ψ satisfies the cusp condition at 21 rr
 ),( 21
Ψ∇ =∑∑
νµµν φφ )()( 21
1 rrc
.                                         ………...(S-2) 
The second derivative ),( 21
Ψ∇  is continuous at 21 rr
=  because each term in 
the expansion is continuous (the rules of continuity for algebraic combinations). 
 This implies that there is no cusp in ),( 21 rr
Ψ  at 21 rr
 BUT ),( 21 rr
Ψ has to satisfy the cusp condition at 21 rr
 This is only possible if ),( 21 rr
Ψ changes the sign at 21 rr
And hence the exact ),( 21 rr
Ψ  has exchange node irrespective of its spin 
multiplicity. 
C.) More illustrative example: 
 Hamiltonian for He atom: 
H +−−∇−∇−=                                                    ………...(S-3) 
 and )2,1()2,1( Ψ=Ψ EH                                                                       ………...(S-4) 
 Let expand  
νµµν φφ )2()1()2,1( c                                                           ………...(S-5) 
Where { })(rµφ  is complete set of eigen functions of the Hamiltonian 
1ˆ 2 −∇−=  with eigenvalue equation )()(ˆ rrh µµµ φεφ = . 
 Rewriting the He Hamiltonian: 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 18 of 21)       
                                                                                                                                                 
H +−−∇−∇−= =
hh ++                            ………...(S-6) 
)2,1(
)2()1()ˆˆ(
)2,1(
)2,1()
)2,1(
)2,1(ˆ
EL +Ψ
νµµν φφ
          ………...(S-7) 
 Since )(rµφ  is an eigen function of ĥ . 
 We can write  
)2,1(
)2()1()(
EL +Ψ
νµνµµν φφεε
                                      ………...(S-8) 
 µννµ εε dE ++=  
 µννµ εε dE −=+                                                                              ………...(S-9) 
)2,1(
)2()1()(
EL +Ψ
νµµνµν φφ
                                   ………...(S-10) 
)2,1(
)2()1(
)2,1(
)2()1(
EL +Ψ
∞ ∞∞ ∞
νµµνµν
νµµν φφφφ
    ………...(S-11) 
)2,1(
)2()1(
)2,1(
)2,1(
EL +Ψ
νµµνµν φφ
                           ………...(S-12) 
)2,1(
)2()1(
EEL +Ψ
νµµνµν φφ
                                      ………...(S-13) 
 If )2,1(Ψ  is exact then 
 EEL =  
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 19 of 21)       
                                                                                                                                                 
 and 
)2,1(
)2()1(
νµµνµν φφ
                                             ………...(S-14) 
Now let assume that the exact solution is symmetric with respect to interchange of 
two electrons. )2,1(Ψ  and )(rµφ  are well behaved and differentiable. From the 
rules of continuity for algebraic combinations, 
the term in equation (S-14), 
)2,1(
)2()1(
νµµνµν φφdc
 is continuous and finite 
and it should not diverge when 012 →r . Therefore, symmetric solution is not 
acceptable. 
However, if )2,1(Ψ  = 0 at r12=0 then 
)2,1(
)2()1(
νµµνµν φφdc
 will also diverge 
and can compensate the divergence in 1/r12 term.   
Thus the only acceptable solution is antisymmetric (with respect to interchange 
of two electrons) solution. 
D.) Another example: 
Almost all QMC people believe (their believe is based on some assumptions and 
approximations) that He atom ground state wave function is symmetric. This is an 
illusion.  This can be understood as follows: 
Let us take trial functions of two electron system: 
( )221
1 xxb
xxg 21
211 ),(
−−=  
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 20 of 21)       
                                                                                                                                                 
( ) eeexx Uxxxxg 21 22212 221),(
( ) eeexx Uxxxxg 21 22213 21),( −
−−=  
If someone claims that He ground state is symmetric, what kind of exact 
symmetric wave functions they are getting finally? The functions g1and g2 are 
symmetric with respect to interchange of two electrons. The functions like g1, g2, 
and g3 can satisfy cusp condition. The functions g1 and g2 are not differentiable at 
x1=x2 and therefore these are not acceptable. The antisymmetric function g3 are 
differentiable at x1=x2. 
However, people have got very accurate ground state energy for He atom using 
HF*Jastrow trial function and they concluded that He atom has no node without 
examining the simultaneous probability of finding two electrons at exactly same 
place. I think, they got good results due to inherent beauty of DMC technique.  
The functions g2 is symmetric and g3 is antisymmetric with respect to interchange 
of two electrons. However, g2*g2 and g3*g3 give the exactly same probability 
distribution i.e. same physics. Functions g2 and g3 vanish when x1=x2. Further, the 
VMC calculation for g2 and g3 will give exactly same energy. Can anyone predict 
that the VMC energies obtained from g2 and g3 represent singlet or triplet state? I 
am sure it is not possible.  
An antisymmetric wave function can satisfy cusp condition as well as it’s 
derivative will be continuous simultaneously at the point of coincidence. Here 
symmetric and antisymmetric wave functions serve same distribution. Why I 
should not prefer antisymmetric wave function for which a boundary condition 
can be imposed easily? 
4. Further, if I assume the argument  “He ground state wave function is 
symmetric and has no such node” is correct. The end will be a nonsense, 
which is as follows: 
____________________________________________________________________________________________________________ 
The Exact Boundary Condition to Solve the Schrödinger Equation of Many Electron System  
By Rajendra Prasad, Village + Post: Kamalpur, District: Chandauli, Uttar Pradesh, PIN: 232106, India E-mail: Theochem@gmail.com 
4/3/2007  (Page 21 of 21)       
                                                                                                                                                 
It is very much common practice in QMC calculation to take Hartree-Fock trial 
function as a product of alpha-beta determinants. For example N atom: 
PSIT(1,2,3,4,5,6,7) = Det(1,2,3,4,5,6,7). 
PSIT(1,2,3,4,5,6,7) = Detα (1,2,3,4,5)*Detβ(6,7). 
PSIT(1,2,3,4,5,6,7) = Detα(1,2,3,4,5)*Detβ(6,7). 
Detα(1,2,3,4,5)*Detβ(6,7) ≠ Detα(1,2,3,4,7)*Detβ(6,5) 
The trial function PSIT is clearly neither symmetric nor antisymmetric with 
respect to interchange of alpha-beta electrons. It is not clear to me what kind of 
final solution (i.e. symmetric or antisymmetric) we will obtain with this trial 
function fixed node DMC? The fact that we can not write PSIexact = Phiα*Phiβ. 
5. It is natural to ask what is node of 3S He atom and why people are getting 
very accurate energy with exchange node r1-r2=0?  
At the moment, I can only say that this is due to artifact of importance sampling 
because people have used HF*Jastrow trial function. The correlation energy for 
He(3S) atom is around 2mH and the overlap of HF trial function with exact wave 
function can be anticipated to be more than 99%. Perhaps due to technical reasons 
final DMC solution may have converged to He(3S). I have seen some recent 
papers on the node of He(3S) system. It is widely claimed that the node r1-r2=0 
belongs to He(3S) system and it is exact. I differ with their argument and I proved 
that the exchange node r1-r2=0 belongs to He ground state. I do not know if 
anyone have performed DMC calculation with a trial function like psit(r1,r2)=(r1-
r2)*exp(-2*r1)*exp(-2*r2) and reported the energy for He(
3S).  Anyway, at present 
I am interested only in the non-degenerate ground state of atoms and molecules. 
Author welcomes further comments, questions, and suggestions if any. 
Theochem@gmail.com
