Reduced Density Matrix
In this chapter we explain reduced density matrix and its applications in molecular modeling. While traditional methods for computing the orbitals are scaling cubically with respect to the number of electrons, the computation of the density matrix offers
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Reduced Density Matrix
10.1 Introduction The solution of an N-body Schrödinger equation through the ground state properties of a fermion system (5.4) in an applied external potential for the analysis of a boundless variety of physical situations remains a focus of research. It was J. E. Mayer in 1955 who first identified that for non-relativistic electrons (which interact via pair forces alone), the system energy depends only upon the two-body reduced density matrix (2-RDM). In fact, only two combinations are possible in this regard; the pair density (2-RDM) and the one-body reduced density matrix (1-RDM). The former one keeps four-particle degrees of freedom while the latter one keeps only two particle degrees of freedom. Mayer suggested the possibility of computing the ground state energy and density matrix information by simply carrying out a Rayleigh-Ritz minimization with respect to the pair density and 1-RDM. However, the initial computations resulted in horrible results due to the ignoring of a number of necessary restrictions or constraints. Progress in this very promising approach could be possible, if and only if we include all the necessary restrictions.
10.2 Reduced Density Matrices The N-fermion problem can be treated as a discrete orthonormal basis of single particle wavefunctions. Let ψ be the ground state normalized wavefunction for an N-fermion system. Hence: ψ |ψ = 1
K. I. Ramachandran et al., Computational Chemistry and Molecular Modeling DOI: 10.1007/978-3-540-77304-7, ©Springer 2008
(10.1)
195
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10 Reduced Density Matrix
1-RDM (γ ) can be defined as: γ (i, i ) = ψ a+ i ai ψ
(10.2)
Here, ai and a+ i are the annihilation and creation operators for the single particle state i for the chosen basis set. An annihilation operator is an operator that lowers the number of particles in a given state by one. A creation operator is an operator that increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. Similarly, 2-RDM (Γ ) is given by: " ! + (10.3) Γ (i, j; i , j ) = ψ a+ a a a i j j i ψ (a j and a+j are the annihilation and creation operators single particle state j for the chosen basis set.). Γ (i, j; i , j ) is antisymmetric under the interchange of i and j and also under the interchange of i and j ; γ and Γ are hermitian. The Hamiltonian of the N-fermion system involving only one-body and twobody interactions can be written as Eq. 10.4: 1 + Hˆ = ∑ h1 (i, i )a+ h2 (i, j, i , j )a+ ∑ i ai + i a j a j ai 2 i,i i, j,i , j
(10.4)
(h1 and h2 are single particle Hamiltonians). The ground state energy E can be expressed exactly in terms of the 1-RDM and 2-RDM: 1 E = Tr(h1 γ ) + Tr(h2Γ ) 2
(10.5)
Tr stands for trace of the operator. Tr(h1 γ ) = ∑ h1 (i, i )γ (i , i)
(10.6)
∑
(10.7)
i,i
Tr(h2Γ ) =
i, j,i , j
h2 (i, j, i , j )Γ (i , j , i, j)
The pair (γ , Γ ) is used as a trial function in the space of functions satisfying the stated antisymmetry and hermiticity conditions. In the computatio
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