CISD

CISD Yes/No A/ transformed integrals n N to solve for one Cl energy and eigenvector... 

Figure B3.1.9 [83] displays the errors (in pieometres eompared to experimental findings) in the equilibrium bond lengths for a series of 28 moleeules obtained at the FIF, MP2-4, CCSD, CCSD(T), and CISD levels of theory using three polarized eorrelation-eonsistent basis sets (valenee DZ tlu-ough to QZ).

Clearly, the HF method, independent of basis, systematically underestimates the bond lengdis over a broad percentage range. The CISD method is neither systematic nor narrowly distributed in its errors, but the MP2 and MP4 (but not MP3) methods are reasonably accurate and have narrow error distributions if valence TZ or QZ bases are used. The CCSD(T), but not the CCSD, method can be quite reliable if valence TZ or QZ bases are used. 

Correlation can be added as a perturbation from the Hartree-Fock wave function. This is called Moller-Plesset perturbation theory. In mapping the HF wave function onto a perturbation theory formulation, HF becomes a hrst-order perturbation. Thus, a minimal amount of correlation is added by using the second-order MP2 method. Third-order (MP3) and fourth-order (MP4) calculations are also common. The accuracy of an MP4 calculation is roughly equivalent to the accuracy of a CISD calculation. MP5 and higher calculations are seldom done due to the high computational cost (A time complexity or worse). 

It is possible to make a method approximately size-extensive by adding a correction to the final energy. This has been most widely used for correcting CISD energies. This is a valuable technique because a simple energy correction formula is easier to work with than full Cl calculations, which require an immense amount of computational resources. The most widely used correction is the Davidson correction ... 

Electron correlation is often very important as well. The presence of multiple bonding interactions, such as pi back bonding, makes coordination compounds more sensitive to correlation than organic compounds. In some cases, the HF wave function does not provide even a qualitatively correct description of the compound. If the weight of the reference determinant in a single-reference CISD calculation is less than about 0.9, then the HF wave function is not qualitatively correct. In such cases, multiple-determinant, MSCSF, CASPT2, or MRCI calculations tend to be the most efficient methods. The alternative is... 

The stmctural parameters of ethylene oxide have been determined by microwave spectroscopy (34). Bond distances iu nm determined are as follows C—C, 0.1466 C—H, 0.1085 and C—O, 0.1431. The HCH bond angle is 116.6°, and the COC angle 61.64°. Recent ah initio studies usiug SCF, MP2, and CISD have predicted bond lengths that are very close to the experimental values (35,36). 

Practical configuration interaction methods augment the Hartree-Fock by adding only a limited set of substitutions, truncating the Cl expansion at some level of substitution. For example, the CIS method adds single excitations to the Hartree-Fock determinant, CID adds double excitations, CISD adds singles and doubles, CISDT adds singles, doubles, and triples, and so on. 

A disadvantage of all these limited Cl variants is that they are not size-consistent.The Quadratic Configuration Interaction (QCI) method was developed to correct this deficiency. The QCISD method adds terms to CISD to restore size consistency. QCISD also accounts for some correlation effects to infinite order. QCISD(T) adds triple substitutions to QCISD, providing even greater accuracy. Similarly, QCISD(TQ) adds both triples and quadruples from the full Cl expansion to QCISD. 

To illustrate the CISD technique, consider dineon (Figure 11.9). HF theory cannot hope to give an accurate description of the dispersion interaction between two neon atoms, so an electron correlation treatment is vital. Here are the results for a separation of 300 pm. 

The HF-LCAO calculation follows the usual lines (Figure 11.10) and the frozen core approximation is invoked by default for the CISD calculation. CISD is iterative, and eventually we arrive at the improved ground-state energy and normalization coefficient (as given by equation 11.7) — Figure 11.11. 

Table 11.1 shows an interesting point about CISD. The energy of the dineon pair at the arbitrarily large separation of 5000 pm is exactly twice the energy of two free atoms at the HF-LCAO level of theory, but this is not the case at the CISD level of theory. We say that HF theory scales correctly, whilst CISD does not. 

The projection equations are then identical with those obtained by minimizing the energy and so the CID and CISD energies are truly variational (they give upper bounds to the full Cl result). 

If we were to use CISD to calculate the expansion coefficients, then the need not be zero they couple in to the ground state through the higher-order excitations. 

As illustrated above, even quite small systems at the CISD level results in millions of CSFs. The variational problem is to extract one or possibly a few of the lowest eigenvalues and -veetors of a matrix the size of millions squared. This cannot be done by standard diagonalization methods where all the eigenvalues are found. There are, however, iterative methods for extraeting one, or a few, eigenvalues and -veetors of a large matrix. The Cl problem eq. (4.6) may be written as... 

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