Why Configuration Interaction?

In the first paper on quantum mechanics, Heisenberg used matrix mechanics to calculate the frequencies and intensities of spectral lines [2]. Later, when Schrödinger discovered wave mechanics, it was quickly shown that the Schrödinger and Heisenberg approaches are mathematically equivalent [3,4]. Given the ease with which matrices may be implemented on a computer, it is entirely natural to attempt to solve the molecular time-independent Schrödinger equation $\hat{H} \Psi = E \Psi$ using matrix mechanics.

Matrix mechanics requires that we choose a vector space for the expansion of the problem. For the case of an N-electron molecule, our wavefunction must be expanded in a basis of N-particle functions (the nuclei need not be considered in the electronic wavefunction, if we have invoked the Born-Oppenheimer approximation). How do we construct the N-particle basis functions? Here we follow the arguments of Szabo and Ostlund [1], p. 60. Assume we have a complete set of functions $\{\chi_i(x_1)\}$ of a single variable x₁. Then any arbitrary function of that variable can be expanded exactly as

$$\Phi(x_1) = \sum_i a_i \chi_i(x_1).$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred1(x_1) = _i a_i _i(x_1). )

How can we expand a function of two variables x₁ and x₂which have the same domain? If we hold x₂ fixed, then

$$\Phi(x_1, x_2) = \sum_i a_i(x_2) \chi_i(x_1).$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred2(x_1, x_2) = _i a_i(x_2) _i(x_1). )

Now note that each expansion coefficient a_i(x₂) is a function of a single variable, which can be expanded as

$$a_i(x_2) = \sum_j b_{ij} \chi_j(x_2).$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred3a_i(x_2) = _j b_ij _j(x_2). )

Substituting this expression into the one for $\Phi(x_1, x_2)$, we now have

$$\Phi(x_1, x_2) = \sum_{ij} b_{ij} \chi_i(x_1) \chi_j(x_2)$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred4 (x_1, x_2) = _ij b_ij _i(x_1) _j(x_2) )

a process which can obviously be extended for $\Phi(x_1, x_2, \ldots, x_N)$.

Let us now collect the spin and space coordinates of an electron into a variable ${\bf x}$. We can write a spin orbital as $\chi({\bf x})$. The result analogous to equation (2.4) for a system of N electrons is

$$\Phi({\bf x}_1, {\bf x}_2, \ldots, {\bf x}_N) = \sum_{ij\ldo... ...\chi_i({\bf x}_1) \chi_j({\bf x}_2) \ldots \chi_N({\bf x}_N)$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred5(x_1, x_2, ..., x_N) = _ij...N b_ij...N _i(x_1) _j(x_2) ... _N(x_N) )

However, the wavefunction must be antisymmetric  with respect to the exchange of the coordinates of any two electrons¹ For the two-particle case, the requirement

$$\Phi({\bf x}_1, {\bf x}_2) = -\Phi({\bf x}_2, {\bf x}_1)$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred6(x_1, x_2) = -(x_2, x_1) )

implies that b_ij = -b_ji and b_ii = 0, or

$$\Phi({\bf x}_1, {\bf x}_2) \sum_{j > i} b_{ij} [ \chi_i({\bf x}_1) \chi_j({\bf x}_2) - \chi_j({\bf x}_1) \chi_i({\bf x}_2) ]$$ =

$$\sum_{j > i} 2^{1/2} b_{ij} \vert\chi_i \chi_j \rangle$$ tex2html_deferredtex2html_deferred2.tex2html_deferred7 = (tex2html_deferredtex2html_deferred2.tex2html_deferred7 & = & _j > i 2^1/2 b_ij |_i _j )

More generally, an arbitrary N-electron wavefunction can be expressed exactly as a linear combination of all possible N-electron Slater determinants formed from a complete set of spin orbitals $\{\chi_i({\bf x})\}$. If we solve the matrix mechanics problem ${\bf H} \vert \Psi \rangle = E \vert \Psi \rangle$in a complete basis of N-electron functions as just described, we will obtain all electronic eigenstates of the system exactly. If our N-electron basis functions are denoted $\vert \Phi_i \rangle$, the eigenvectors of ${\bf H}$ are given as

$$\vert \Psi_j \rangle = \sum_{i}^{I} c_{ij} \vert \Phi_i \rangle$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred8| _j = _i^I c_ij | _i )

if there are I possible N-electron basis functions (I will be infinite if we actually have a complete set of one electron functions $\chi_i$). The matrix ${\bf H}$ is constructed so that $H_{ij} = \langle \Phi_i \vert H \vert \Phi_j \rangle$ for $i,j = 1,2,\ldots, I$. The matrix elements H_ij may be written in terms of one- and two-electron integrals according to ``Slater's rules,''  as discussed in section 2.4.

The N-electron basis functions $\vert \Phi_i \rangle$ can be written as substitutions or ``excitations'' from the Hartree-Fock ``reference'' determinant, i.e.

$$\vert \Psi \rangle = c_0 \vert \Phi_0 \rangle + \sum_{ra} c^r... ...t,a<b<c} c^{rst}_{abc} \vert \Phi^{rst}_{abc} \rangle + \ldots$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred9| = c_0 | _0 + _ra c^r_a | ^r_a + _a<b,r<s c^rs_ab | ^rs_ab + _r<s<t,a<b<c c^rst_abc | ^rst_abc + ...  )

where $\vert \Phi_a^r \rangle$ means the Slater determinant formed by replacing spin-orbital a in $\vert \Phi_0 \rangle$ with spin orbital r, etc. Every N-electron Slater determinant can be described by the set of N spin orbitals from which it is formed, and this set of orbital occupancies is often referred to as a ``configuration.'' Thus the ``configuration interaction'' method is, in its most straigtforward implementation, nothing more or less than the matrix mechanics solution of the time-independent non-relativistic electronic Schrödinger equation $\hat{H} \Psi = E \Psi$. One of the great strengths of the CI method is its generality; the formalism applies to excited states, to open-shell systems, and to systems far from their equilibrium geometries. By contrast, traditional single-reference perturbation theory  and coupled-cluster  approaches generally assume that the reference configuration is dominant, and they may fail when it is not.

In practice, one does not have a complete set of one-particle basis functions $\{\chi_i({\bf x})\}$; typically one assumes that the incomplete one-electron basis set is large enough to give useful results, and the CI procedure is not modified. The quality of the one-particle basis set can be checked by comparing the results of calculations using progressively larger basis sets.

It is also possible to reduce the size of the N-electron basis set. If we desire only wavefunctions of a given spin and/or spatial symmetry, as is usually the case, we need include only those N-electron basis functions of that symmetry, since the Hamiltonian matrix is block-diagonal according to space and spin symmetries. This point is discussed further in section 4.1. If one performs the matrix mechanics calculation using a given set of one-particle functions $\{\chi_i({\bf x})\}$ and all possible N-electron basis functions $\{\vert \Phi_i \rangle\}$ (possibly symmetry-restricted), the procedure is called ``full CI.''    The full CI corresponds to solving Schrödinger's equation exactly within the space spanned by the specified one-electron basis. If the one-electron basis is complete (it never is in practice, but it may be in theory), then the procedure is called a ``complete CI''  [5].

Unfortunately, even with an incomplete one-electron basis, a full CI  is computationally intractable for any but the smallest systems, due to the vast number of N-electron basis functions required (the size of the CI space is discussed in section 4.4). The CI space must be reduced somehow--hopefully in such a way that the approximate CI wavefunction and energy are as close as possible to the exact values. The effective reduction of the CI space is a major concern in CI theory, and we will discuss some of the more popular approaches in these notes.

By far the most common CI approximation is the truncation of the CI space expansion according to excitation level relative to the reference state (equation 2.9). The widely-employed CI singles and doubles (CISD)  wavefunction includes only those N-electron basis functions which represent single or double excitations relative to the reference state. Since the Hamiltonian operator includes only one- and two-electron terms, only singly and doubly excited configurations can interact directly with the reference, and they typically account for about 95% of the correlation energy   in small molecules at their equilibrium geometries [6]. Truncation of the CI space according to excitation class is discussed more thoroughly in section 4.2.