Two-particle excitation

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

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 ... 

This section summarises some properties common to non-polar stoichiometric oxide surfaces and presents theoretical arguments to explain them. Their specificities come from the local environment of the surface sites, which have a lower coordination number than in the bulk. From this point of view, a close parallel with unsupported clusters or ultra-thin films can be established [3]. We will not explicitely consider here the properties associated to structural defects, such as steps or kinks, for the reason of space limitation. However, most of the time, the same concepts as those akin to terrace sites apply, but with an even larger strength since the local environment is more reduced. We will successively analyse structural characteristics, energetics, electron distribution, one-particle and two-particle excitations. 

This part takes account for dynamic correlation between the test particles and the finite system and is necessary for the complete description of two particle excitation processes. Of course, also here the unphysical components of the extended states enter. 

Table 10 Calculated energies of one- and two-particle excitations in NiO crystal field splitting Acp(eV), Racah B parameter (eV), and analytically predicted value for the two-particle excitation ...

Acf and AExz,/z z2,x2-/ = 3B- -2Acf. We extract the Racah B parameter from the two single-particle excitations, and use the derived value to calculate the energy of the two-particle excitation, as a check on the consistency of the mapping as can be seen in Table 10, the agreement is excellent. The free ion value for B is most closely matched by a weight of 50%, but this is purely due to the difference between excitation energies a better match with their absolute placement is obtained by uncorrelated UHF calculations. We may further conclude that density functional theory is unable to captme the small but subtle correlation corrections involved in d—>d excitations, at least not in NiO. 

The energy of a two-particle excitation is higher at the saddle point than that in the stably deformed configuration. Thus, it is sufficient to consider only collective excited states as transition states. 

II. THE GREEN S FUNCTIONS, SINGLE- AND TWO-PARTICLE EXCITATIONS, AND THEIR INTERRELATION WITH DENSITY-FUNCTIONAL THEORY... 

A method of increments [111, 159,160] is a wavefunction-based ab-initio correlation method for solids. This method is closely related to the ideas of the local ansatz (LA), [5] where local operators acting on the SCF wavefunction are used to admix suitable one- and two-particle excitations to the mean-field HF ground state. The many-electron Hamiltonian is split according to (5.44) and the ground-state Hamiltonian Hscf and the corresponding wavefunction scf = are assumed to be known. 

The following operators are dehned Ai, where i should be considered as a compact index that includes the bond i as weU as the one- and two-particle excitations of bond i, and Aij, which describes the two-particle excitations where one excitation is out of bond i while the other is out of bond j. Within the restricted operator subspace spanned by Ai and A j the operator Q can be written in the form... 

To complete the development of this operator formalism, we can define the two-particle excitation operators. 

The operator with si = 2 = 1 is identical in form to the nonrelativistic two-particle excitation operator. To arrive at expressions for the time-reversed operators, any index in this expression can be replaced with the barred index. The final expression for the Dirac-Coulomb Hamiltonian is... 

These operators can be made symmetric under time reversal by multiplying the E operators by i however, this makes it more difficult to define the two-particle excitation operators e. 

This expression is more compact than the equivalent expression in terms of the one-and two-particle excitation operators E and e. 

The matrix elements between the CSFs in the real basis involve two Hamiltonian matrix elements between determinants, Hpq and HpQ. However, HpQ is only nonzero for Mk P)-Mk Q) < 2, because otherwise the excitation between P and Q is more than a two-particle excitation. Thus it is only in the center blocks of the Hamiltonian that the linear combination of matrix elements needs to be taken. 

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