Nodal surfaces in Quantum Monte Carlo: a user’s guide Dario Bressanini, Gabriele Morosi, Silvia Tarasco

Dipartimento di Scienze Chimiche e Ambientali, Università dell’Insubria, Como

FIXED-NODE DIFFUSION MONTE CARLO

In order to study fermionic systems with Diffusion Monte Carlo, antisymmetry properties of the wavefunction are usually imposed by means of the fixed-node approximation (FN-DMC), that is one adopts the nodal surface of a trial wavefunction, assuming that is a good approximation of the exact one. If, during the simulation, a walker crosses the node of the trial function, the move is rejected.

+ -

If the trial wavefunction’s nodes are the exact ones, FN-DMC gives the exact energy; otherwise, calculated energy is affected by the so called nodal error.

The aim of this work is to find a way to improve the nodes of trial wavefunctions systematically , reducing nodal error.

He

2 +

The Hartree Fock wavefunction of the molecular ion He 2 +     ( 1 )  ( 2 )   ( 2 ) has the simple form:  ( 1 )  1 2 1 2 Nodes only depend on the coordinates of two electrons, i.e. on six variables. Fixing some of these coordinates it is possible to plot cuts of the nodal surface so that one can see how FN-DMC energy (hence the quality of the nodal surface) changes when different trial wave functions are used.

• Basis set

DIMERS



Li

2

n. det

1 9 3 11 5

E

exact

Added CSF

2

σ g

2

nσ g

2 1  2

ux

 1  2

uy n

 2

ux



n

 2

uy

2

p z



u

2  3

p z

 2

g

E(E

h

) % Corr

-14.9919(1) 97.2(1)% -14.9907(2) 96.7(1)% -14.9933(1) 98.3(1)% -14.9933(1) 98.3(1)% -14.9952(1) 99.8(1)% -14.9954

Only configurations built with orbitals of different angular momentum and symmetry contribute to the shape of the nodes 

C

2

Using the same basis set, it is possible to select the CSFs that contribute to the construction of the exact nodal surface.

Orbitals built with different ab

initio methods give different nodal

surfaces. Correlated methods give better nodes.

CSF E(E

h

)

1 [4] -75.8600(12) 36 [4] -75.9025(7) 1 a 1 b 12 b -75.8622(2) -75.8694(3) -75.9032(8)

E

exact

-75.9192 [4] a. Hartree Fock orbitals b. CI-NO orbitals

% Corr

88.5(3)% 96.7(1)% 88.89(4)% 90.29(6)% 96.9(1)%

CONCLUSIONS

 As the basis set increases, the wavefunction improves. This is not always true for nodes.

 Orbitals built with the same basis set but with different ab

initio methods have different nodes. Correlated methods give

functions with better nodal surfaces.

 In multideterminantal wavefunctions, some determinants perturb the nodal surface. A cut-off criterion on the linear coefficient of the determinants is not the right criterion in the selection of CSFs.



A better knowledge about wavefunction nodes will improve the precision of our DMC results and reduce the computational cost of calculations.

1s E= -4.9905(2) E j 2(1s) E= -4.9927(1) E h 4(1s) E= -4.9940(1) E h 5(1s) [1] E= -4.9926(1) E h  As the basis set increases (from SZ to 4Z) FN-DMC energy improves and the nodal surface’s curvature decreases. If one adds another 1s function, nodes get worse.

The function built with the s basis set gives the exact energy within the statistical error.

When this basis set is augmented with diffuse functions (sp basis set), energy increases and so nodes get worse.

A

b initio method

2(1s)1(2s)1(3s) (s) E= -4.9943(2) E h

These are exact nodes

S2(1s)1(2s)1(3s)2(2p) [2] E= -4.9932(1) E h

E

exact

=-4.994598 E

h

[3]

CAS E= -4.9939(2) E j CI-NO E= -4.9918(2) E j 

Multideterminantal expansion

In order to improve the quality of ab

initio wavefunctions, multideterminantal

expansions are normally used.

With the SZ basis set, the three determinants wavefunction has really worse nodes than the Hartree Fock one.

On the other hand, the same expansion made with the sp basis gives an energy improvement: surface.

the two added determinants contribute positively to the construction of the exact nodal Within the same basis, it is also important the ab initio method one uses, The CAS wavefunction has better nodes then the Hartree Fock wavefunction calculated with the same basis set. On the contrary, CI-NO wavefunction has nodes with a large curvature, very different from the exact ones.

1s (3 DET) E= -4.9778(3) E j sp (3 DET) E= -4.9946(1) E h

References:

[ 1] E.Clementi and C.Roetti, At. Data and Nucl. Data Tables 14,177 (1974).

[2] P.N.Regan, J.C.Brown and F.A.Matsen, Phys. Rev. 132,304(1963).

[3] W.Cenceck and J.Rychlewski J. Chem. Phys 102, 6, (1995).

[4] R.N. Barnett, Z.Sun and W.A.Lester. J. Chem. Phys. 114, 2013 (2001).