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Nodal surfaces in Quantum Monte Carlo a users
guide
Dario Bressanini, Gabriele Morosi, Silvia
Tarasco Dipartimento di Scienze Chimiche e
Ambientali, Università dellInsubria, Como

• He2
• The Hartree Fock wavefunction of the molecular
  ion He2 has the simple form
• 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

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. The aim of this work is to find
a way to improve the nodes of trial wavefunctions
systematically , reducing nodal error.
If the trial wavefunctions nodes are the exact
ones, FN-DMC gives the exact energy otherwise,
calculated energy is affected by the so called
nodal error.
1s E -4.9905(2) Ej
2(1s) E -4.9927(1) Eh
4(1s) E -4.9940(1) Eh
5(1s) 1 E -4.9926(1) Eh

• DIMERS
• Li2
• C2

As the basis set increases (from SZ to 4Z) FN-DMC
energy improves and the nodal surfaces 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.
Only configurations built with orbitals of
different angular momentum and symmetry
contribute to the shape of the nodes
2(1s)1(2s)1(3s) (s) E -4.9943(2) Eh
S2(1s)1(2s)1(3s)2(2p) 2 E -4.9932(1) Eh
These are exact nodes Eexact-4.994598 Eh 3
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.
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.
CI-NO E -4.9918(2) Ej
CAS E -4.9939(2) Ej
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.
a. Hartree Fock orbitals b. CI-NO orbitals

• 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 (3 DET) E -4.9778(3) Ej
sp (3 DET) E -4.9946(1) Eh
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).