Operator excitation

This may be further transformed by an inner projection onto a complete set of excitation and de-excitation operators, h this is equivalent to inserting a resolution of the identity in the operator space (remember that superoperators work on operators). 

So far everything is exact. A complete manifold of excitation operators, however, means that all excited states are considered, i.e. a full Cl approach. Approximate versions of propagator methods may be generated by restricting the excitation level, i.e. tmncating h. A complete specification furthermore requires a selection of the reference, normally taken as either an HF or MCSCF wave function. 

The one-particle excitation operator T and the two-particle excitation operator T2 are defined by ... 

At each excitation level beyond the single-excitation level, a number of terms contribute. For example, double excitations are generated both by means of the double-excitation operator T2 (connected excitations)... 

Here H is the unperturbed Hamiltonian of the system and h denotes a complete set of excitation and de-excitation operators, arranged as column vector h or as row vector h. Completeness of the set of operators h means that all possible excited states of the system must be generated by operating on... 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

Each sector excitation operator is, in the usual way, a sum of virtual excitations of... 

C. Spin-Free Excitation Operators and -Particle Density Matrices... 

In our formalism [5-9] excitation operators play a central role. Let an orthonormal basis p of spin orbitals be given. This basis has usually a finite dimension d, but it should be chosen such that in the limit —> cxd it becomes complete (in the so-called first Sobolev space [10]). We start from creation and annihilation operators for the ij/p in the usual way, but we use a tensor notation, in which subscripts refer to annihilation and superscripts to creation ... 

The Kronecker delta is written here in a tensor notation. One can define excitation operators as normal products (or products in normal order) of the same number of creation and annihilation operators normal order in the original sense means that all creation operators have to be on the left of all annihilation operators). 

These operators are particle-number conserving, that is, action of any excitation operator on an -electron wavefunction (with n arbitrary) leads again to an -electron wavefunction (or deletes it). 

In order to define excitation operators, one need not start from the creation and annihilation operators one can instead simply require that action of, for example, aP on a Slater determinant >1) with occupied and (for p q) ij/p unoccupied replaces by ij/p. Otherwise it annihilates >1). 

Any product of two or more excitation operators can be written as a sum of excitation operators, for example. 

Most Hamiltonians of physical interest are spin-free. Then the matrix elements in Eq. (9) depend only on the space part of the spin orbitals and vanish for different spin by integration over the spin part. Then it is recommended to eliminate the spin and to deal with spin-free operators only. We start with a basis of spin-free orbitals cpp, from which we construct the spin orbitals orbital labels (capital letters) and spin labels... 

Greek letters). We define spin-free excitation operators carrying only orbital labels, by summation over spin... 

The spin-free two-particle excitation operators and density matrices are symmetric with respect to simultaneous exchange of the upper and lower indices, but neither symmetric nor antisymmetric with respect to exchange of either upper or lower indices separately ... 

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