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
He2
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 Hartree Fock wavefunction of the molecular ion He2+ has the simple form:
  1 (1)2 (2)  1 (2)2 (1)
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
The aim of this work is to find a way to improve the nodes of trial
wavefunctions systematically , reducing nodal error.
DIMERS
Added CSF
1
2σ g2
-14.9919(1)
97.2(1)%
9
nσ
2
g
-14.9907(2)
96.7(1)%
3
11
1  1
-14.9933(1)
-14.9933(1)
98.3(1)%
98.3(1)%
5
2 p z  u2  3 p z  2g -14.9952(1)
99.8(1)%
Eexact
2
ux
2
uy
2
2
nux
 nuy
E(Eh)
2(1s)
4(1s)
5(1s) [1]
E= -4.9905(2) Ej
E= -4.9927(1) Eh
E= -4.9940(1) Eh
E= -4.9926(1) Eh
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.
 Li2
n. det
1s
% Corr
Only configurations built
with
orbitals
of
different
angular
momentum and symmetry
contribute to the shape
of the nodes
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.
2(1s)1(2s)1(3s) (s)
S2(1s)1(2s)1(3s)2(2p) [2]
E= -4.9943(2) Eh
E= -4.9932(1) Eh
These are exact nodes
Eexact=-4.994598 Eh [3]
 Ab initio method
-14.9954
Within the same basis, it is also
important the ab initio method one uses,
 C2
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(Eh)
1 [4] -75.8600(12)
36 [4] -75.9025(7)
1a
1b
12b
-75.8622(2)
-75.8694(3)
-75.9032(8)
% Corr
88.5(3)%
96.7(1)%
CAS
88.89(4)%
90.29(6)%
96.9(1)%
E= -4.9939(2) Ej
Eexact -75.9192 [4]
a. Hartree Fock orbitals
b. CI-NO orbitals
CONCLUSIONS
 As the basis set increases, the wavefunction improves. This is
not always true for nodes.
CI-NO
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.
E= -4.9918(2) Ej
 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:
the
two
added
determinants contribute positively to
the construction of the exact nodal
surface.
1s (3 DET)
sp (3 DET)
E= -4.9778(3) Ej
E= -4.9946(1) Eh
 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.
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).