Reduced Hamiltonian operator

Since 1 have assumed that ij/ is normalized to 1, the trace of is also 1 and the trace of P2 is N N — 1). We now define the reduced Hamiltonian operator... 

Here QHQ defines a reduced Hamiltonian operator II. On substituting a formal solution of the second set of equations into the first, it follows for rj — 0+ that... 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

Of course, the Coulomb interaction appears in the Hamiltonian operator, H, and is often invoked for interpreting the chemical bond. However, the wave function, l7, must be antisymmetric, i.e., must satisfy the Pauli exclusion principle, and it is the only fact which explains the Lewis model of an electron pair. It is known that all the information is contained in the square of the wave function, 1I7 2, but it is in general much complicated to be analyzed as such because it depends on too many variables. However, there have been some attempts [3]. Lennard-Jones [4] proposed to look at a quantity which should keep the chemical significance and nevertheless reduce the dimensionality. This simpler quantity is the reduced second-order density matrix... 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

The formal solution of Al-representability for the 2-RDM is developed in terms of a convex set of two-particle reduced Hamiltonian matrices. To complement the derivation of the positivity conditions from the metric matrices, we derive them from classes of these two-particle reduced Hamiltonian matrices. This interpretation allows us to demonstrate that the 2-positivity conditions are exact for certain classes of Hamiltonian operators for any interaction strength. In this section all of the ROMs are normalized to unity. Much of this discussion appeared originally in Refs. [21, 29]. 

A. J. Coleman and 1. Absar, Reduced Hamiltonian orbitals. 111. Unitarily invariant decomposition of Hermitian Operators. Int. J. Quantum Chem. 18, 1279 (1980). 

As in the previous section, by connected we mean all terms that scale linearly with N. Wedge products of cumulant RDMs can scale linearly if and only if they are connected by the indices of a matrix that scales linearly with N transvec-tion). In the previous section we only considered the indices of the one-particle identity matrix in the contraction (or number) operator. In the CSE we have the two-particle reduced Hamiltonian matrix, which is defined in Eqs. (2) and (3). Even though the one-electron part of scales as N, the division by A — 1 in Eq. (3) causes it to scale linearly with N. Hence, from our definition of connected, which only requires the matrix to scale linearly with N, the transvection... 

The space of linear operators on Fq is equal to the space of possible g-body reduced Hamiltonians, Hq. ... 

Recall that in his Theorems 3 and 4 Hans Kummer [3] defined a contraction operator, L, which maps a linear operator on A-space onto an operator on p-space and an expansion operator, E, which maps an operator on p-space onto an operator on A-space. Note that the contraction and expansion operators are super operators in the sense that they act not on spaces of wavefunctions but on linear spaces consisting of linear operators on wavefunction spaces. If the two-particle reduced Hamiltonian is defined as... 

A similar equation is obtained for the reduced density operator T8 = trp[r] for the s-region. The right-hand side can not be written in general as a commutator between a hamiltonian and a density operator, so that in this case... 

The theory is based on an optimized reference state that is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions (r). This is the simplest form of the more general orbital functional theory (OFT) for an iV-electron system. The energy functional E = (4> // < >)is required to be stationary, subject to the orbital orthonormality constraint (i j) = Sij, imposed by introducing a matrix of Lagrange multipliers kj,. The general OEL equations derived above reduce to the UHF equations if correlation energy Ec and the implied correlation potential vc are omitted. The effective Hamiltonian operator is... 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

This approximation is numerically stable since it is unitary. The method is suitable for short time dynamics and time-dependent Hamiltonian operators. To reduce... 

Starting from the gap Hamiltonian (33) and the interactions with the reservoirs (34) and (35), eliminating the reservoir degrees of freedom within the standard Bom-Markov approach we derive a master equation for the reduced density operator of the atomic ensemble. The calculation is lengthy but straight forward. Disregarding level shifts caused by the bath interaction we find for the populations in states cj ) the following density matrix equations in the interaction picture ... 

One promising approach to the problem of effectively reducing the number of direct Cl matrix-vector products is the approximate Hamiltonian operator method of Werner and coworkers - described in Section III. This is an extended micro-iterative method in which the Hamiltonian operator is allowed to be approximated during the solution of the wavefunction correction vector within an MCSCF iteration. 

In this section, we go beyond the perturbative treatment and calculate the eigenstates of large excitonic systems by diagonalizing the Hamiltonian operator on an efficient reduced basis set [164]. 

Enhancement of two-photon cross-sections by two-dimensional and three-dimensional arrangements of monomers has been demonstrated with fluorene V-shapes and dendrimeric structures. Such multidimensional structures lead to lower two-photon absorptions than Hnear oUgomers, but they have better one-photon transparencies. An accurate calculation of large exitonic systems is obtained by diagonalizing the Hamiltonian operator on a reduced basis set. 

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