The problem of the motion of a particle attracted by two fixed centers of force according to the Coulomb force law can be treated by classical mechanics and has been used in quantum theory by Pauli and Messen for a theory of the hydrogen molecule ion.1 In the quantum mechanics, where the energy levels are determined as the “eigenwerte” of Schrodinger’s equation, the variables are separable and the boundary value problem is easily set up. But thus far a satisfactory treatment of the differential equations involved is lacking. Burrau2 has recently carried out a numerical integration of the problem for the lowest energy level of an electron moving under the influence of two fixed centers of Coulomb attraction as a function of the distance apart of these centers. In this paper, Burrau’s data are used to give a semi-quantitative discussion of the neutral hydrogen molecule. His values are:
Nuclear separation
1.0
1.3
1.6
1.8
2.0
2.2
2.4
2.95
Electronic energy
2.896
2.648
2.436
2.309
2.204
2.109
2.025
1.836
The unit of separation is the Bohr l1 orbit radius of hydrogen atom, that of energy is the ionization potential of atomic hydrogen.
The authors tested several approximations commonly used in molecular energy calculations by applying them to the six-site half-filled Hubbard model and comparing the results with the exact solution. Three main conclusions follow from this study: (i) for $\frac{\ensuremath{\beta}}{U}$ in the range of interest for molecules, perturbations to the molecular-orbital ground state produce better results than the other more elaborate methods tested; (ii) Nesbet's pair theory is only slightly worse in this range but has the advantage of working well for all values of the parameters, and (iii) perturbative corrections to the unrestricted Hartree-Fock ground state give very poor results when this state has lower symmetry than the Hamiltonian.
The wave functions for the hydrogen atom are determined in the momentum representation, using momentum variables
pr, ${p}_{\ensuremath{\theta}}$, ${p}_{\ensuremath{\varphi}}$ which are associated with the spherical polar variables
r, $t(={e}^{i\ensuremath{\pi}cos\ensuremath{\theta}})$ and $v({e}^{i\ensuremath{\varphi}})$. The functions may be written as ${{\ensuremath{\varphi}}_{n,l,m}({p}_{r},{p}_{\ensuremath{\theta}},{p}_{\ensuremath{\varphi}})={\ensuremath{\alpha}}_{n,l}({p}_{r}){\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})\ensuremath{\delta}({p}_{\ensuremath{\varphi}}+m),}{{\ensuremath{\alpha}}_{n,l}({p}_{r})={\left(\frac{n}{i{p}_{0}}\right)}^{3l+z}\ensuremath{\Sigma}\stackrel{n\ensuremath{-}l\ensuremath{-}1}{k=0}\left({n\ensuremath{-}l\ensuremath{-}1}{k}\right)\frac{{2}^{k\ensuremath{-}l+1}}{(2l+k+1)(2l+k)\ensuremath{\cdots}(l+k+2)}{\left(\frac{\frac{i{p}_{0}}{n}}{{p}_{r}\ensuremath{-}\frac{i{p}_{0}}{n}}\right)}^{l+k+2},}{{\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})=\ensuremath{\Sigma}\stackrel{E(\frac{l}{2})}{j=E(\frac{m}{2})}{{a}_{j}{p}_{\ensuremath{\theta}}}^{\ensuremath{-}\frac{(m+1)}{2}}{J}_{\frac{j\ensuremath{-}(m\ensuremath{-}1)}{2}}({p}_{\ensuremath{\theta}}),}$ where $\ensuremath{\delta}$ is the Dirac delta function and ${p}_{0}=Z{a}_{0}^{\ensuremath{-}1}$ (
a0 is the Bohr radius). ${J}_{\ensuremath{\nu}}({p}_{0})$ are Bessel functions of integer or half-integer order. The radial functions are shown to have poles up to order
n+1 at
pr=inp0.
Total energies of charge and spin states of a specifically designed target molecule 2,5,8-trihydro-phenalenyl (
3H-C13H9 from here on), calculated by means of multiconfigurational self-consistent-field (MCSCF) and density-functional-theory-B3LYP methods, have been fitted by means of interacting Pariser-Parr-Pople (PPP) (intrasite and intersite Coulomb interactions) and Hubbard Hamiltonians for $\ensuremath{\pi}$ electrons. Numerically exact many-body solutions for these models were obtained by a variant of the Lanczos algorithm. Our calculation shows that the combination MCSCF-PPP with a hopping integral $t=\ensuremath{-}2.63\text{ }\text{eV}$ and a local Coulomb repulsion
U=10.51 eV produces the best results. Both model parameters are close to values frequently used in the literature, sometimes just for the Hubbard model. The fit of MCSCF energies by the Hubbard model is not as satisfactory (root-mean-square deviation is larger and $t=\ensuremath{-}4.83\text{ }\text{eV}$ and
U=21.27 eV parameters are far from common values). Neither of both models is able to accurately reproduce B3LYP energies. On the other hand, the value of the ratio
∣U/t∣ is always close to 4 indicating the proximity to a magnetic phase transition in extended systems. Additionally, we find that Lieb's theorem for the Hubbard model on bipartite lattices applies to neutral
3H-C13H9 since all ab initio methods and model Hamiltonians predict a fivefold spin-degenerate ground state.
The dimerization of polyacetylene is studied using a distance-dependent Hubbard Hamiltonian for the valence $\ensuremath{\pi}$ electrons and approximating $\ensuremath{\sigma}$-electron contributions by a sum of pair interactions ${E}_{\ensuremath{\sigma}}{(r}_{\mathrm{ij}})$. The $\ensuremath{\pi}$-electron Coulomb repulsion
U is taken to be independent of the C-C distances
rij and is varied systematically from the uncorrelated or H\"uckel limit all over to the strongly correlated or Heisenberg regime. For each value of
U, ${E}_{\ensuremath{\sigma}}{(r}_{\mathrm{ij}})$, and the $\ensuremath{\pi}$-electron hopping integral
t(rij) are fitted to reproduce ab initio results on the ground-state and first-excited-state energy of the ethylene molecule. The one-dimensional many-body problem is solved numerically by means of the density-matrix renormalization-group (DMRG) method. The ground-state energy
E of the infinite open polymer is obtained by extrapolation considering chains of up to 300 atoms. The dimerization $\ensuremath{\delta}{=(r}_{i,i+1}\ensuremath{-}{r}_{i\ensuremath{-}1,i})/2$ and the average bond length ${R=(r}_{i,i+1}{+r}_{i\ensuremath{-}1,i})/2$ are determined by minimizing
E. While
R is not significantly affected by the value of
U, $\ensuremath{\delta}$ shows a remarkable nonmonotonic behavior. The differences between the delocalized regime, small
U/t, and spin-Peierls behavior, large
U/t, are discussed. The convergence properties of the DMRG algorithm are analyzed as a function of
U, $\ensuremath{\delta}$, and the number states kept at each RG step. Criteria are examined in order to control the accuracy of the DMRG calculations in both weakly and strongly correlated situations.
Abstract In this article, the hydrogen molecular ground-state energies using our algorithm based on quantum variational principle are calculated. They are calculated through a simulator since the system of the present study (i.e., the hydrogen molecule) is relatively small and hence the ground-state energies for this molecule are efficiently classically simulable using a simulator. Complete details of this algorithm are elucidated. For this, a full description on the fermions–qubits and the molecular Hamiltonian–qubit Hamiltonian transformations, is given. The authors search for qubit system parameters ( θ 0 {\theta }_{0} and θ 1 {\theta }_{1} ) that yield the minimum energies for the system and also study the ground state energies as a function of the molecular bond length. Proposed circuit is humble and does not include many parameters compared with that of Kandala et al., the authors control only two parameters ( θ 0 {\theta }_{0} and θ 1 {\theta }_{1} ).