Two alternative conditions for the stationarity of the energy expectation value with respect to k-particle excitations are the k-particle Brillouin conditions BCk and the k-particle contracted Schrödinger equations, CSEk. These conditions express the k-particle density matrices γk in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the γk in terms of their cumulants λk, but this is not sufficient to make the BCk or CSEk separable (extensive), i.e., they are not expressible in terms of only connected diagrams. However, in a formulation based on the recently introduced general normal ordering with respect to arbitrary wave functions, the irreducible counterparts IBCk and ICSEk of the BCk and CSEk can be defined. They are easily evaluated explicitly in terms of the generalized Wick theorem for arbitrary wave functions, and they lead to equations for the direct construction of the cumulants λk, which are additively separable quantities and which scale linearly with the system size. The IBCk or the ICSEk are necessary conditions for γ and the λk to represent an exact n-fermionic eigenstate of the given Hamiltonian. To specify the desired state, additional conditions must be satisfied as well, e.g., the partial trace relations which relate λ2 to γ and γ2. The particle number and the total spin must be specified and n-representability conditions enter implicitly. While the nondiagonal elements of γ and the λk are determined by the IBCk or the ICSEk, the additional conditions mainly serve to fix the diagonal elements. A hierarchy of k-particle approximations is defined. It is based on the fact that the expansion in terms of cumulants λk can be truncated at any particle rank, which would not be possible for the density matrices γk. For closed-shell states the one-particle approximation agrees with Hartree–Fock.
Two alternative conditions for the stationarity of the energy expectation value with respect to k-particle excitations are the k-particle Brillouin conditions BCk and the k-particle contracted Schrödinger equations, CSEk. These conditions express the k-particle density matrices γk in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the γk in terms of their cumulants λk, but this is not sufficient to make the BCk or CSEk separable (extensive), i.e., they are not expressible in terms of only connected diagrams. However, in a formulation based on the recently introduced general normal ordering with respect to arbitrary wave functions, the irreducible counterparts IBCk and ICSEk of the BCk and CSEk can be defined. They are easily evaluated explicitly in terms of the generalized Wick theorem for arbitrary wave functions, and they lead to equations for the direct construction of the cumulants λk, which are additively separable quantities and which scale linearly with the system size. The IBCk or the ICSEk are necessary conditions for γ and the λk to represent an exact n-fermionic eigenstate of the given Hamiltonian. To specify the desired state, additional conditions must be satisfied as well, e.g., the partial trace relations which relate λ2 to γ and γ2. The particle number and the total spin must be specified and n-representability conditions enter implicitly. While the nondiagonal elements of γ and the λk are determined by the IBCk or the ICSEk, the additional conditions mainly serve to fix the diagonal elements. A hierarchy of k-particle approximations is defined. It is based on the fact that the expansion in terms of cumulants λk can be truncated at any particle rank, which would not be possible for the density matrices γk. For closed-shell states the one-particle approximation agrees with Hartree–Fock.
Recently [W. Kutzelnigg and D. Mukherjee, Chem. Phys. Lett. 317, 567 (2000); D. Mukherjee and W. Kutzelnigg, J. Chem. Phys 114, 2047 (2001)] the irreducible k-particle Brillouin conditions IBCk and the irreducible k-particle contracted Schrödinger equations ICSEk were derived. These permit the definition of a hierarchy of k-particle approximations for the direct calculation of the cumulants λk of the k-particle density matrices. Now, the spin-free form of these conditions, appropriate for a spin-free Hamiltonian, is given. This is particularly useful for open-shell states. The definition of the cumulants of the reduced densities has to be generalized for these anyway, making use of irreducible tensor operators with respect to SU2. There are two alternative definitions of spin-free cumulants, of which the one in terms of spin-free reduced density matrices appears to be preferable. Alternatively to the straight spin-free formulation, we also present a theory in terms of spin-free operators adapted to the symmetric group. Partial trace relations that relate the elements of the cumulants of different particle rank are derived. There are partial trace relations for “exchange elements,” which are determined by the total spin quantum number S. From these relations the individual exchange elements of Λ2 can be obtained in special cases. This allows a simple formulation of the stationarity conditions for open-shell states of any spin multiplicity.
The k-particle irreducible Brillouin conditions IBCk and the k-particle irreducible contracted Schrödinger equations ICSEk for a closed-shell state are analyzed in terms of a Møller-Plesset-type perturbation expansion. The zeroth order is Hartree-Fock. From the IBC2(1), i.e., from the two-particle IBC to first order in the perturbation parameter mu, one gets the leading correction lambda2(1) to the two-particle cumulant lambda2 correctly. However, in order to construct the second-order energy E(2), one also needs the second-order diagonal correction gammaD(2) to the one-particle density matrix gamma. This can be obtained: (i) from the idempotency of the n-particle density matrix, i.e., essentially from the requirement of n-representability; (ii) from the ICSE1(2); or (iii) by means of perturbation theory via a unitary transformation in Fock space. Method (ii) is very unsatisfactory, because one must first solve the ICSE3(2) to get lambda3(2), which is needed in the ICSE2(2) to get lambda2(2), which, in turn, is needed in the ICSE1(2) to get gamma(2). Generally the (k+1)-particle approximation is needed to obtain Ek correctly. One gains something, if one replaces the standard hierarchy, in which one solves the ICSEk, ignoring lambda(k+1) and lambda(k+2), by a renormalized hierarchy, in which only lambda(k+2) is ignored, and lambda(k+1) is expressed in terms of the lambdap of lower particle rank via the partial trace relation for lambda(k+2). Then the k-particle approximation is needed to obtain E(k) correctly. This is still poorer than coupled-cluster theory, where the k-particle approximation yields E(k+1). We also study the possibility to use some simple necessary n-representability conditions, based on the non-negativity of gamma(2) and two related matrices, in order to get estimates for gammaD(2) in terms of lambda2(1). In general these estimates are rather weak, but they can become close to the best possible bounds in special situations characterized by a very sparse structure of lambda2 in terms of a localized representation. The perturbative analysis does not encourage the use of a k-particle hierarchy based on the ICSEk (or on their reducible counterparts, the CSEk), it rather favors the approach in terms of the unitary transformation, where the k-particle approximation yields the energy correct up to E(2k-1). The problems that arise are related to the unavoidable appearance of exclusion-principle violating cumulants. The good experience with perturbation theory in terms of a unitary transformation suggests that one should abandon a linearly convergent iteration scheme based on the ICSEk hierarchy, in favor of a quadratically convergent one based on successive unitary transformations.
We analyze the structure and the solutions of the irreducible k-particle Brillouin conditions (IBCk) and the irreducible contracted Schrödinger equations (ICSEk) for an n-electron system without electron interaction. This exercise is very instructive in that it gives one both the perspective and the strategies to be followed in applying the IBC and ICSE to physically realistic systems with electron interaction. The IBC1 leads to a Liouville equation for the one-particle density matrix gamma1=gamma, consistent with our earlier analysis that the IBC1 holds both for a pure and an ensemble state. The IBC1 or the ICSE1 must be solved subject to the constraints imposed by the n-representability condition, which is particularly simple for gamma. For a closed-shell state gamma is idempotent, i.e., all natural spin orbitals (NSO's) have occupation numbers 0 or 1, and all cumulants lambdak with k> or =2 vanish. For open-shell states there are NSO's with fractional occupation number, and at the same time nonvanishing elements of lambda2, which are related to spin and symmetry coupling. It is often useful to describe an open-shell state by a totally symmetric ensemble state. If one wants to treat a one-particle perturbation by means of perturbation theory, this mainly as a run-up for the study of a two-particle perturbation, one is faced with the problem that the perturbation expansion of the Liouville equation gives information only on the nondiagonal elements (in a basis of the unperturbed states) of gamma. There are essentially three possibilities to construct the diagonal elements of gamma: (i) to consider the perturbation expansion of the characteristic polynomial of gamma, especially the idempotency for closed-shell states, (ii) to rely on the ICSE1, which (at variance with the IBC1) also gives information on the diagonal elements, though not in a very efficient manner, and (iii) to formulate the perturbation theory in terms of a unitary transformation in Fock space. The latter is particularly powerful, especially, when one wishes to study realistic Hamiltonians with a two-body interaction.
k -particle cumulants λk (for 2⩽k⩽n) corresponding to the k-particle reduced density matrices γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ≡γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3−3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3−3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration self-consistent field wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk corresponding to the spin-free reduced density matrices Γk are also defined and analyzed. The main interest in the density cumulants is in connection with the recently formulated normal ordering and the corresponding Wick theorem for arbitrary reference functions, but they are also useful for an analysis of electron correlation.
A generalization of normal ordering and of Wick’s theorem with respect to an arbitrary reference function Φ as some generalized “physical vacuum” is formulated in a different (but essentially equivalent) way than that suggested previously by one of the present authors. Guiding principles are that normal order operators with respect to any reference state must be expressible as linear combinations of those with respect to the genuine vacuum, that the vacuum expectation value of a normal order operator must vanish (with respect to the vacuum to which it is in normal order), and that the well-known formalism for a single Slater determinant as physical vacuum must be contained as a special case. The derivation is largely based on the concepts of “Quantum Chemistry in Fock space,” which means that particle-number-conserving operators (excitation operators) play a central role. Nevertheless, the contraction rules in the frame of a generalized Wick theorem are derived, that hold for non-particle-number-conserving operators as well. The contraction rules are formulated and illustrated in terms of diagrams. The contractions involve the “residual n-particle density matrices” λ, which are the irreducible (non-factorizable) parts of the conventional n-particle density matrices γ, in the sense of a cumulant expansion for the density. A spinfree formulation is presented as well. The expression of the Hamiltonian in normal order with respect to a multiconfiguration reference function leads to a natural definition of a generalized Fock operator. MC-SCF-theory is easily worked out in this context. The paper concludes with a discussion of the excited configurations and the first-order interacting space, that underlies a perturbative coupled cluster type correction to the MCSCF function for an arbitrary reference function, and with general implications of the new formalism, that is related to “internally contracted multireference configuration interaction.” The present generalization of normal ordering is not only valid for arbitrary reference functions, but also if the reference state is an ensemble state.