89 research outputs found
An efficient implementation of Slater-Condon rules
Slater-Condon rules are at the heart of any quantum chemistry method as they
allow to simplify -dimensional integrals as sums of 3- or 6-dimensional
integrals. In this paper, we propose an efficient implementation of those rules
in order to identify very rapidly which integrals are involved in a matrix
element expressed in the determinant basis set. This implementation takes
advantage of the bit manipulation instructions on x86 architectures that were
introduced in 2008 with the SSE4.2 instruction set. Finding which spin-orbitals
are involved in the calculation of a matrix element doesn't depend on the
number of electrons of the system.Comment: 8 pages, 5 figure
Fixed-Node Diffusion Monte Carlo potential energy curve of the fluorine molecule F2 using selected configuration interaction trial wavefunctions
The potential energy curve of the F molecule is calculated with
Fixed-Node Diffusion Monte Carlo (FN-DMC) using Configuration Interaction
(CI)-type trial wavefunctions. To keep the number of determinants reasonable
(the first and second derivatives of the trial wavefunction need to be
calculated at each step of FN-DMC), the CI expansion is restricted to those
determinants that contribute the most to the total energy. The selection of the
determinants is made using the so-called CIPSI approach (Configuration
Interaction using a Perturbative Selection made Iteratively). Quite remarkably,
the nodes of CIPSI wavefunctions are found to be systematically improved when
increasing the number of selected determinants. To reduce the non-parallelism
error of the potential energy curve a scheme based on the use of a
-dependent number of determinants is introduced. Numerical results show that
improved FN-DMC energy curves for the F molecule are obtained when
employing CIPSI trial wavefunctions. Using the Dunning's cc-pVDZ basis set the
FN-DMC energy curve is of a quality similar to that obtained with FCI/cc-pVQZ.
A key advantage of using selected CI in FN-DMC is the possibility of improving
nodes in a systematic and automatic way without resorting to a preliminary
multi-parameter stochastic optimization of the trial wavefunction performed at
the Variational Monte Carlo level as usually done in FN-DMC.Comment: 16 pages, 15 figure
Quantum Monte Carlo with very large multideterminant wavefunctions
An algorithm to compute efficiently the first two derivatives of (very) large
multideterminant wavefunctions for quantum Monte Carlo calculations is
presented. The calculation of determinants and their derivatives is performed
using the Sherman-Morrison formula for updating the inverse Slater matrix. An
improved implementation based on the reduction of the number of column
substitutions and on a very efficient implementation of the calculation of the
scalar products involved is presented. It is emphasized that multideterminant
expansions contain in general a large number of identical spin-specific
determinants: for typical configuration interaction-type wavefunctions the
number of unique spin-specific determinants
() with a non-negligible weight in the expansion is
of order . We show that a careful implementation
of the calculation of the -dependent contributions can make this
step negligible enough so that in practice the algorithm scales as the total
number of unique spin-specific determinants, , over a wide range of total number of determinants (here,
up to about one million), thus greatly reducing the total
computational cost. Finally, a new truncation scheme for the multideterminant
expansion is proposed so that larger expansions can be considered without
increasing the computational time. The algorithm is illustrated with
all-electron Fixed-Node Diffusion Monte Carlo calculations of the total energy
of the chlorine atom. Calculations using a trial wavefunction including about
750 000 determinants with a computational increase of 400 compared to a
single-determinant calculation are shown to be feasible.Comment: 9 pages, 3 figure
Spin density distribution in open-shell transition metal systems: A comparative post-Hartree-Fock, Density Functional Theory and quantum Monte Carlo study of the CuCl2 molecule
We present a comparative study of the spatial distribution of the spin
density (SD) of the ground state of CuCl2 using Density Functional Theory
(DFT), quantum Monte Carlo (QMC), and post-Hartree-Fock wavefunction theory
(WFT). A number of studies have shown that an accurate description of the
electronic structure of the lowest-lying states of this molecule is
particularly challenging due to the interplay between the strong dynamical
correlation effects in the 3d shell of the copper atom and the delocalization
of the 3d hole over the chlorine atoms. It is shown here that qualitatively
different results for SD are obtained from these various quantum-chemical
approaches. At the DFT level, the spin density distribution is directly related
to the amount of Hartree-Fock exchange introduced in hybrid functionals. At the
QMC level, Fixed-node Diffusion Monte Carlo (FN-DMC) results for SD are
strongly dependent on the nodal structure of the trial wavefunction employed
(here, Hartree-Fock or Kohn-Sham with a particular amount of HF exchange) : in
the case of this open-shell system, the 3N -dimensional nodes are mainly
determined by the 3-dimensional nodes of the singly occupied molecular orbital.
Regarding wavefunction approaches, HF and CASSCF lead to strongly localized
spin density on the copper atom, in sharp contrast with DFT. To get a more
reliable description and shed some light on the connections between the various
theoretical descriptions, Full CI-type (FCI) calculations are performed. To
make them feasible for this case a perturbatively selected CI approach
generating multi-determinantal expansions of reasonable size and a small
tractable basis set are employed. Although semi-quantitative, these near-FCI
calculations allow to clarify how the spin density distribution evolves upon
inclusion of dynamic correlation effects. A plausible scenario about the nature
of the SD is proposed.Comment: 13 pages, 12 Figure
Alternative definition of excitation amplitudes in Multi-Reference state-specific Coupled Cluster
A central difficulty of state-specific Multi-Reference Coupled Cluster
(MR-CC) formalisms concerns the definition of the amplitudes of the single and
double excitation operators appearing in the exponential wave operator. If the
reference space is a complete active space (CAS) the number of these amplitudes
is larger than the number of singly and doubly excited determinants on which
one may project the eigenequation, and one must impose additional conditions.
The present work first defines a state-specific reference-independent operator
which acting on the CAS component of the wave function
maximizes the overlap between
and the eigenvector of the CAS-SD CI
matrix . This operator may be used to generate
approximate coefficients of the Triples and Quadruples, and a dressing of the
CAS-SD CI matrix, according to the intermediate Hamiltonian formalism. The
process may be iterated to convergence. As a refinement towards a strict
Coupled Cluster formalism, one may exploit reference-independent amplitudes
provided by to define a
reference-dependent operator by fitting the eigenvector of the
(dressed) CAS-SD CI matrix. The two variants, which are internally
uncontracted, give rather similar results. The new MR-CC version has been
tested on the ground state potential energy curves of 6 molecules (up to
triple-bond breaking) and a two excited states. The non-parallelism error with
respect to the Full-CI curves is of the order of 1 m.Comment: 11 page
Curing basis-set convergence of wave-function theory using density-functional theory: a systematically improvable approach
The present work proposes to use density-functional theory (DFT) to correct
for the basis-set error of wave-function theory (WFT). One of the key ideas
developed here is to define a range-separation parameter which automatically
adapts to a given basis set. The derivation of the exact equations are based on
the Levy-Lieb formulation of DFT, which helps us to define a complementary
functional which corrects uniquely for the basis-set error of WFT. The coupling
of DFT and WFT is done through the definition of a real-space representation of
the electron-electron Coulomb operator projected in a one-particle basis set.
Such an effective interaction has the particularity to coincide with the exact
electron-electron interaction in the limit of a complete basis set, and to be
finite at the electron-electron coalescence point when the basis set is
incomplete. The non-diverging character of the effective interaction allows one
to define a mapping with the long-range interaction used in the context of
range-separated DFT and to design practical approximations for the unknown
complementary functional. Here, a local-density approximation is proposed for
both full-configuration-interaction (FCI) and selected
configuration-interaction approaches. Our theory is numerically tested to
compute total energies and ionization potentials for a series of atomic
systems. The results clearly show that the DFT correction drastically improves
the basis-set convergence of both the total energies and the energy
differences. For instance, a sub kcal/mol accuracy is obtained from the
aug-cc-pVTZ basis set with the method proposed here when an aug-cc-pV5Z basis
set barely reaches such a level of accuracy at the near FCI level
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