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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}}} .
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 ...
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
where is a molecular orbital represented as the sum of n atomic orbitals , each multiplied by a corresponding coefficient , and r (numbered 1 to n) represents which atomic orbital is combined in the term. The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital.
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).
For instance, the lone pairs of water are usually treated as two equivalent sp x hybrid orbitals, while the corresponding "nonbonding" orbitals of carbenes are generally treated as a filled σ(out) orbital and an unfilled pure p orbital, even though the lone pairs of water could be described analogously by filled σ(out) and p orbitals (for ...
The resulting equations are a set of non-linear equations, which are solved in an iterative manner. Standard quantum-chemistry packages (GAMESS (US), NWChem, ACES II, etc.) solve the coupled-cluster equations using the Jacobi method and direct inversion of the iterative subspace extrapolation of the t-amplitudes to accelerate convergence.