Truncated configuration interaction

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. 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

There is also a hierarchy of electron correlation procedures. The Hartree-Fock (HF) approximation neglects correlation of electrons with antiparallel spins. Increasing levels of accuracy of electron correlation treatment are achieved by Mpller-Plesset perturbation theory truncated at the second (MP2), third (MP3), or fourth (MP4) order. Further inclusion of electron correlation is achieved by methods such as quadratic configuration interaction with single, double, and (perturbatively calculated) triple excitations [QCISD(T)], and by the analogous coupled cluster theory [CCSD(T)] [8],... 

The electron correlation problem remains a central research area for quantum chemists, as its solution would provide the exact energies for arbitrary systems. Today there exist many procedures for calculating the electron correlation energy (/), none of which, unfortunately, is both robust and computationally inexpensive. Configuration interaction (Cl) methods provide a conceptually simple route to correlation energies and a full Cl calculation will provide exact energies but only at prohibitive computational cost as it scales factorially with the number of basis functions, N. Truncated Cl methods such as CISD (A cost) are more computationally feasible but can still only be used for small systems and are neither size consistent nor size extensive. Coupled cluster... 

Thus the one-particle basis determines the MOs, which in turn determine the JV-particle basis. If the one-paxticle basis were complete, it would at least in principle be possible to form a complete jV-particle basis, and hence to obtain an exact wave function variationally. This wave function is sometimes referred to as the complete Cl wave function. However, a complete one-paxticle basis would be of infinite dimension, so the one-paxticle basis must be truncated in practical applications. In that case, the iV-particle basis will necessarily be incomplete, but if all possible iV-paxticle basis functions axe included we have a full Cl wave function. Unfortunately, the factorial dependence of the iV-paxticle basis size on the one-particle basis size makes most full Cl calculations impracticably large. We must therefore commonly use truncated jV-paxticle spaces that axe constructed from truncated one-paxticle spaces. These two truncations, JV-particle and one-particle, are the most important sources of uncertainty in quantum chemical calculations, and it is with these approximations that we shall be mostly concerned in this course. We conclude this section by pointing out that while the analysis so fax has involved a configuration-interaction approach to solving Eq. 1.2, the same iV-particle and one-particle space truncation problems arise in non-vaxiational methods, as will be discussed in detail in subsequent chapters. 

It is appropriate at this point to compare some formal properties of the three general approaches to dynamical correlation that we have introduced configuration interaction, perturbation theory, and the coupled-cluster approach. First, we note that taken fax enough (all degrees of excitation in Cl and CC, infinite order of perturbation theory) all three approaches will give the same answer. Indeed, in a complete one-paxticle basis all three will then give the exact answer. We axe concerned in this section with the properties of truncated Cl and CC methods and finite-order perturbation theory. 

Truncated Cl Wavefunctions.—Between the limits of a minimal and a full configuration interaction one is faced with the problem of choosing how many configurations and more importantly, which configurations are to be included. A computational problem arises because of the very slow convergence, generally... 

The correspond to different electron configurations. In configuration interaction o is the Hartree-Fock function (or an approximation to it in a truncated basis set) and the other 4>t are constructed from virtual orbitals which are the by-product of the Hartree-Fock calculation. The coefficients Ci are found by the linear-variation method. Unfortunately, the so constructed are usually an inadequate basis for the part of the wavefunction not represented by [Pg.5]

The exponential operator T creates excitations from 4>o according to T = l + 72 + 73 + , where the subscript indicates the excitation level (single, double, triple, etc.). This excitation level can be truncated. If excitations up to Tn (where N is the number of electrons) were included, vPcc would become equivalent to the full configuration interaction wave function. One does not normally approach this limit, but higher excitations are included at lower levels of coupled-cluster calculations, so that convergence towards the full Cl limit is faster than for MP calculations. 

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