Lowdin partition technique

Following Lowdin [32] we assume that we know a subspace of vectors which contains a good approximation of the exact ground state vector. In addition we can think that the low-energy excited states of interest also belong to this same subspace. Then let P be the projection operator onto this subspace and Q = I — P be the complementary projection operator satisfying the following conditions  

These are nothing but the conditions of orthogonality of the subspace of interest lm/ (lm/ - image P - stands here for the set of vectors of a linear space which are obtained by action of the linear operator P upon all vectors of the linear vector space) and its complementary subspace IrriQ. 

To apply the partition of the whole vector space to the solution of the Schrodinger equation with the exact Hamiltonian II h hi V we multiply this equation from the left in turn by P and Q and making use of the fact that 

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P. O. Lowdin. Partitioning technique, perturbation theory, and rational approximations. Intern. J. Quantum Chem., 21 69, 1982. 

A central problem in physics and chemistry has always been the solution of the Schrodinger equation (SE) for stationary states. Such stationary states may relate to electronic structure problems, in which case one is primarily interested in bound states, or to scattering problems, in which case the stationary solutions are continuum states. In both cases, one of the most powerful tools in the theoretical arsenal for solving such problems is the partitioning technique (PT), which has been developed in a series of papers prominently by Per-Olov Lowdin [1-6] and Herman Feshbach [7-9]. 

The partitioning technique developed by Lowdin provides a synthesis... 

These challenges can be dealt with the powerful mathematical tools of quantum chemistry, as advocated by Per-Olov Lowdin.[l, 2, 3, 4] In our studies, linear algebras with matrices,[4] partitioning techniques,[3] operators and superoperators in Liouville space, and the Liouville-von Neumann... 

Partitioning technique refers to the division of data into isolated sections and it was put into successful practice in connection with matrix operations. Lowdin, in his pioneering studies, [21, 22] developed standard finite dimensional formulas into general operator transformations, including treatments appropriate for both the bound state and the continuous part of the spectrum, see also details in later appendices. Complementary generalizations to resonance-type problems were initiated in Ref. [23], and simple variational formulations were demonstrated in Refs. [24,25]. Note that analogous forms were derived for the Liouville equation [26, 27] and further developed in connection with a retarded-advanced subdynamics formulation [28]. 

Studies in Perturbation Theory. I. An Elementary Iteration-Variation Procedure for Solving the Schrodinger Equation by Partitioning Technique P.-O. Lowdin... 

There are several ways of reducing the one-electron relativistic equation (11.2.7) to a form involving the Pauli spin operators. The method we use here is due to Lowdin (1964) and utilizes the matrix partitioning technique of Section 2.5. 

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