Orbital occupation numbers

Table 4.4 Natural orbital occupation numbers for the distorted acetylene model in Figure 4.11. Only the occupation numbers for the six central orbitals are shown...

The orbital occupation numbers n, (eigenvalues of the density matrix) will be between 0 and 1, corresponding to the number of electrons in the orbital. Note that the representation of the exact density normally will require an infinite number of natural orbitals. The first N occupation numbers N being the total number of electrons in the system) will noraially be close to 1, and tire remaining close to 0. 

The choice of the active spaces for the MCSCF calculations on HF was based on the natural orbital occupancy numbers obtained in MP2 calculations... 

One of the first consequences of the above ideas was the development of the Orbital Local Plasma Approximation (OLPA) by Meltzer et al. [37-39]. The main ingredients in the OLPA consist in approximating the orbital weight factors by the orbital occupation numbers and adapting the Lindhard-Scharff Local Plasma Approximation (LPA) [10-12] to an orbital scheme whereby the orbital mean excitation energy was originally defined as [37,38]... 

A CAS, (6331) or (94), which was chosen according to the MP2 natural orbital occupation numbers [71]. They turn out to be slightly larger than a CAS which includes two correlating orbitals for each HF occupied orbital, (6330) or (93). 

Up to now we have been discussing the local properties of the exchange-correlation potential as a function of the spatial coordinate r. However there are also important proi rtira of the exchange-correlation potential as a function of the particle number. In fact there are close connections between the properties as a function of the particle number and the local properties of the exchange-correlation potential. For instance the bumps in the exchange-correlation potential are closely related to the discontinuity properties of the potential as a function of the orbital occupation number [38]. For heteronuclear diatomic molecules for example there are also similar connections between the bond midpoint shape of the potential and the behavior of the potential as a function of the number of electrons transferred from one atomic fragment to another when... 

The 1-matrix can be diagonalized and its eigenfunctions are the natural orbitals. Equation (41) then implies that the natural orbital occupation numbers he between zero and one, inclusive. Except for the normalization condition. 

The ensemble search in Eq. (82) is the Kohn-Sham procedure, generalized to allow fractional orbital occupation numbers [55, 57-59]. Equation (82) can... 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

For systems with more than one electron pair, the simple picture illustrated above obviously breaks down. The approximate validity of the independent electron-pair model, however, still makes it possible to estimate different correlation effects also in many-electron systems from an inspection of the natural orbital occupation numbers. 

CV and valence calculations leads to almost insuperable linear dependence problems in the resulting basis set [97], We should note that these problems axe perhaps at their worst for the first row. For heavier elements the separation between the valence shell and the core (meaning the next innermost shell here) is not as great, so the disparities in correlating orbital occupation number are less. Suitable ANO sets can often be obtained by correlating all the desired electrons [48, 98]. 

The Janak s theorem (eq.17) and the hardness tensor definition (eq. 20) allows the calculations of Tiij as the first derivative of the Kohn-Sham orbital eigenvalues with respect to the orbital occupation numbers [17] ... 

The relation to the density matrix elements in the spin-orbital occupation numbers representation recovers from noticing that the rows of indices of spin-orbitals ki,k2,..., / y = K (defining a row of creation operators. ..a a , forming a basis Slater determinant) can be in the same manner considered as a set of electronic coordinates in the spin-orbital representation as is the list xi, X2,. .., xjv, ... 

The resolvent in eq. (1.208) is called the one-electron Green s function and the notation for it reads G (z). The integration contour may be set in such a way that it encloses all the poles of the resolvent corresponding to the occupied MOs giving by this the required total projection operator. In the spin-orbital occupation number and the second quantization representations related to each other, one can write the operator projecting to the occupied (spin)-MO as an operator of the number of particles in it. Indeed, the expression... 

Matrix elements associated with the Coulomb integrals 7w. These are expressed in terms of the orbital occupation number operators. Since is an eigenfunction of Epp we get... 

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