Configuration-interaction calculations

To get tme Hartree-Fock orbitals, an infinite set of basis functions should be included in the expansion (2.14). The question is How can we improve our 

Configuration interaction (Cl) is conceptually the simplest procedure for improving on the Hartree-Fock approximation. Consider the determinant formed from the n lowest-energy occupied spin orbitals this determinant is o) and represents the appropriate SCF reference state. In addition, consider the determinants formed by promoting one electron from an orbital k to an orbital v that is unoccupied in these are the singly excited determinants ). Similarly, consider doubly excited (k, v,t) determinants and so on up to n-tuply excited determinants. Then use these many-electron wavefimctions in an expansion describing the Cl many-electron wavefunction Fo)  

Equation (2.18) is a linear variation function. (The summation indices prevent double-counting of excited configurations.) The expansion coefficients cq, c, c%, and so on are varied to minimize the variational integral. I o) is a better approximation than l o)- In principle, if the basis were complete. Cl would provide an exact solution. Here we use a truncated expansion retaining only determinants D that differ from I Tq) by at most two spin orbitals this is a singly-doubly excited Cl (SDCI). 

The presence of excited determinants in 4 o) introduces integrals of the type D AD dr, where A is an operator. Following the Condon-Slater mles, the n-electron integrals can be reduced to sums of one- and two-electron integrals [49]. Consider two determinants D and D, written as in (2.17), arranged so that as many as possible of their left-hand columns match. A one-electron operator ft (viz., —jVj or —Zjrd introduces the new integral 

D differ by two or more spin orbitals. The two-electron operator l/ri2 introduces 

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HyperChem always com putes the electron ic properties for the molecule as the last step of a geometry optimization or molecular dyn am ics calcu lation. However, if you would like to perform a configuration interaction calculation at the optimized geometry, an additional sin gle poin t calcu lation is requ ired with theCI option being turned on. 

The configuration interaction calculation with all possible excitations is called a full Cl. The full Cl calculation using an infinitely large basis set will give an exact quantum mechanical result. However, full Cl calculations are very rarely done due to the immense amount of computer power required. 

A configuration interaction calculation uses molecular orbitals that have been optimized typically with a Hartree-Fock (FIF) calculation. Generalized valence bond (GVB) and multi-configuration self-consistent field (MCSCF) calculations can also be used as a starting point for a configuration interaction calculation. 

A configuration interaction calculation is available only for single points when the reference ground state is obtained from an RHF calculation. 

The amount of computation for MP2 is determined by the partial transformation of the two-electron integrals, what can be done in a time proportionally to m (m is the number of basis functions), which is comparable to computations involved in one step of CID (doubly-excited configuration interaction) calculation. To save some computer time and space, the core orbitals are frequently omitted from MP calculations. For more details on perturbation theory please see A. Szabo and N. Ostlund, Modem Quantum Chemistry, Macmillan, New York, 1985. 

A CASSCF calculation is a combination of an SCF computation with a full Configuration Interaction calculation involving a subset of the orbitals. The orbitals involved in the Cl are known as the active space. In this way, the CASSCF method optimizes the orbitals appropriately for the excited state. In contrast, the Cl-Singles method uses SCF orbitals for the excited state. Since Hartree-Fock orbitals are biased toward the ground state, a CASSCF description of the excited state electronic configuration is often an improvement. 

The next step might be to perform a configuration interaction calculation, in order to get a more accurate representation of the excited states. We touched on this for dihydrogen in an earlier chapter. To do this, we take linear combinations of the 10 states given above, and solve a 10 x 10 matrix eigenvalue problem to find the expansion coefficients. The diagonal elements of the Hamiltonian matrix are given above (equation 8.7), and it turns out that there is a simplification. 

To taike advantage of procedures used for configuration interaction calculations, eigenvalues of the symmetrized matrices, H -I- H, are computed. 

Configuration interaction calculations (3, 4) indicate a possible existence of systems belonging to the second group of our classification (triplets). Although these systems have an even number of electrons and no degenerate... 

The asymptotic energy values obtained by a configuration interaction calculation at 25 a.u. corrected by the coulombic repulsion term (the l/R" term has been neglected) are seen to be in quite good agreement with experiment (Table 3). 

Vilkas, M.J., Ishikawa, Y. and Koc, K. (1998) Quadratically convergent multiconfiguration Dirac-Fock and multireference relativistic configuration-interaction calculations for many-electron systems. Physical Review E, 58, 5096-5110. 

Ab initio DDCI2 (difference-dedicated configuration interaction) calculations seem to provide accurate estimations of the magnetic exchange coupling constants, as demonstrated for the doubly azido-bridged nickel(II) dimer [Ni2(terpy)2(/i-l,l-N3)2]2+.2133... 

The configuration interaction calculations using the PNA singlet excited states (4.2, 4 37, 4.38, 5.57,... 

Dunning basis sets have been optimized with atomic configuration interaction calculations and show steady improvement as the basis set quality is increased. The cc-pVQZ set is the most accurate in this category, but its size probably will preclude its use in the larger calculations... 

For approximate wavefunctions, however, the various formulations give rise to different theoretical predictions. This has been demonstrated in detail, for example, by Hush and Williams (31) for large aromatic systems. Thus, when we wish to obtain exact values of J, we must be very careful in deciding which formalism to use. A final point here is that the one-electron model does not take into account configuration interaction. Calculations for relatively simple systems would be useful here. 

Here we examine the carbon net charges of ethane and ethylene, obtained from SCF and configuration interaction calculations, corrected by means of the appropriate p, determined for n = —4.4122. Remember that the same value of p applies to both ethane and ethylene, as n is solely determined by the effectiveness of the inductive effects. Equation (5.15) is used to get p, namely, p = 138.68 me in 4-31G + Cl calculations and thus, from Eq. (5.10), the corresponding carbon charges of ethane and ethylene (see Table 5.5). 

Basis sets for use in practical Hartree-Fock, density functional, Moller-Plesset and configuration interaction calculations make use of Gaussian-type functions. Gaussian functions are closely related to exponential functions, which are of the form of exact solutions to the one-electron hydrogen atom, and comprise a polynomial in the Cartesian coordinates (x, y, z) followed by an exponential in r. Several series of Gaussian basis sets now have received widespread use and are thoroughly documented. A summary of all electron basis sets available in Spartan is provided in Table 3-1. Except for STO-3G and 3 -21G, any of these basis sets can be supplemented with additional polarization functions and/or with diffuse functions. It should be noted that minimal (STO-3G) and split-valence (3-2IG) basis sets, which lack polarization functions, are unsuitable for use with correlated models, in particular density functional, configuration interaction and Moller-Plesset models. Discussion is provided in Section II. 

This relation is found to agree with results from experiments and configuration interaction calculations (Esbjerg and Nprskov, 1980). 

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