Direct Methods for Solving the Schrödinger Equation

Abstract

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 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.