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Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be exactly solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals. [2]
The variational theorem guarantees that the lowest value of that gives rise to a nontrivial (that is, not all zero) solution vector (,, …,) represents the best LCAO approximation of the energy of the most stable π orbital; higher values of with nontrivial solution vectors represent reasonable estimates of the energies of the remaining π ...
The Slater-type orbital (STO) is a form without radial nodes but decays from the nucleus as does a hydrogen-like orbital. The form of the Gaussian type orbital (Gaussians) has no radial nodes and decays as e − α r 2 {\displaystyle e^{-\alpha r^{2}}} .
It is a special case of the configuration interaction method in which all Slater determinants (or configuration state functions, CSFs) of the proper symmetry are included in the variational procedure (i.e., all Slater determinants obtained by exciting all possible electrons to all possible virtual orbitals, orbitals which are unoccupied in the electronic ground state configuration).
The orbital wave functions are positive in the red regions and negative in the blue. The right column shows virtual MO's which are empty in the ground state, but may be occupied in excited states. In chemistry, a molecular orbital (/ ɒr b ə d l /) is a mathematical function describing the location and wave-like behavior of an electron in a ...
Some quantum chemistry software uses sets of Slater-type functions (STF) analogous to Slater type orbitals, but with variable exponents chosen to minimize the total molecular energy (rather than by Slater's rules as above). The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions (which ...
The solution and are called molecular orbital and orbital energy respectively. Although Hartree-Fock equation appears in the form of a eigenvalue problem, the Fock operator itself depends on ϕ {\displaystyle \phi } and must be solved by a different technique.
The possible orbital symmetries are listed in the table below. For example, an orbital of B 1 symmetry (called a b 1 orbital with a small b since it is a one-electron function) is multiplied by -1 under the symmetry operations C 2 (rotation about the 2-fold rotation axis) and σ v '(yz) (reflection in the molecular