Mostafanejad, Mohammad
Description
The notions of electron correlation and correlation problem
arising in the framework of approximate solutions to the
Schrödinger equation are presented. Then, we briefly review the
original ideas of explicit inclusion of the interelectronic
distance, r12, into the wavefunction as a solution to this
problem.
Exemplifying the efficiency of the explicit correlation for
achieving high accuracy, we analyze the Nakatsuji's
free-complement (FC) method. We...[Show more] demonstrate that at each FC
order, fewer number of complement functions is required to get
lower energies compared with those resulting from the
conventional FC method. Applying the FC method to the triplet
excited state of the He atom, we have discovered the appearance
of permanents in addition to the determinants in the FC expansion
of the wavefunction. These permanents are shown to be important
for the energy convergence.
To achieve a better understanding about the explicitly correlated
methods, especially, the R12 and F12 methods, we analyzed three
possible candidates with various correlation functions F(r_{12})
for a compact and efficient ansatz. Our main focus on the linear
correlation factor r12 has led this analysis to the
investigation of the correlated molecular orbital (CMO) theory of
the Frost and Braunstein (FB). We revisit CMO theory within both
restricted (R) and unrestricted formalisms (U). We also introduce
the unrestricted FB (UFB) ansatz for the first time and derive
the necessary expressions for both RFB and UFB overlap, kinetic,
nuclear-attraction and interelectronic Coulomb repulsion matrix
elements. All integrals have been obtained in closed form except
one for which, we have used an accurate one-dimensional
quadrature.
Finally, we investigate the potential energy curve (PEC) of UFB
for H2 at small, intermediate and large internuclear distances.
Then, we compare its performance with that of RFB, restricted
Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and
configuration interaction (CI) wavefunctions. Reproducing the RFB
results for a much wider range of bond lengths in H2 reveals that
the calculations of FB contain significant errors. We have also
found a pole in the RFB linear correlation coefficient. Our UFB
ansatz provides significant improvement over the RFB where
passing the symmetry breaking point it completely removes the
hump in the RFB PEC. The UFB ansatz also shows surprising
features such as the presence of multiple solutions, non-smooth
PEC, symmetry-broken solutions that are higher in energy than the
restricted solution and RFB->UFB stability in the presence of
lower UFB solutions. These phenomena can have significant impacts
on the explicitly correlated calculations such as R12 and F12
within the unrestricted framework. Also, a detailed discussion on
the large-$R$ asymptotic analysis of these five wavefunctions
shows that none of these PECs has the correct R^{-6} decay within
the minimal basis model. The UFB energy, however, demonstrates
dispersion-like O(R^{-8}) decay which is an improvement over the
CI and UHF with exponential decays. Considering the generalized
FB (GFB) wavefunction where r12^n is the correlation factor and
$n$ is a positive integer, we have shown that no analytic
function of r12 can capture the dispersion within the minimal
basis.
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