| Configuration Interaction |
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In contrast to the Hartree-Fock method, in order to account for Electron Correlation , CI uses a variational wave function that is a linear combination of Configuration State Function s built from spin orbitals (denoted by the superscript ''SO''), : where Ψ is usually the electronic ground state of the system. If the expansion includes all possible CSF s of the appropriate symmetry, then this is a Full Configuration Interaction procedure which exactly solves the electronic Schrödinger Equation within the space spanned by the one-particle basis set. The first term in the above expansion is normally the Hartree-Fock determinant. The other CSFs can be characterised by the number of spin orbitals that are swapped with virtual orbitals from the Hartree-Fock determinant. If only one spin orbital differs, we describe this as a single excitation determinant. If two spin orbitals differ it is a double excitation determinant and so on. This is used to limit the number of determinants in the expansion. For example, the method CID is limited to double excitations only. The method CISD is limited to single and double excitations. Single excitations on their own do not mix with the Hartree-Fock determinant. These methods, CID and CISD, are in many standard programs. The Davidson Correction can be used to estimate a correction to the CISD energy to account for higher excitations. When solving the CI equations, approximations to excited states are also obtained, which differ in the values of their coefficients ''cI''. The CI procedure leads to a General Matrix Eigenvalue Equation : : where ''c'' is the coefficient vector, '''''e''''' is the eigenvalue matrix, and the elements of the hamiltonian and overlap matrices are, respectively, | ||
| :<math> \mathbb{S} {ij} | \left\langle \Phi_i^{SO} \Phi_j^{SO}
ight
angle </math> |
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| Slater Determinants Are Constructed From Sets Of Orthonormal Spin Orbitals, So That <math>\left\langle \Phi I^{SO} \Phi J^{SO} Ight Angle | \delta_{ij}</math>, making <math>\mathbb{S}</math> the identity matrix and simplifying the above matrix equation |
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