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

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). 

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. 

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. 

Figure 5-1. The natural orbital occupation numbers for the N2 molecule as a function of bond distance for the DZ FCI calculation...

Table 25.1 Natural orbital occupation numbers for the ozone molecule with two different active spaces.

In Tables XVI and XVII we have collected the natural orbital occupation numbers from different CASSCF calculations on a selection of octahedral and tetrahedral molecules of first-row transition metals. In all cases a CASSCF calculation was performed using a basic active space of 10 orbitals. For some of the molecules additional calculations with larger active spaces (up to 14 orbitals) are also presented. Only ground-state results are presented, except for the molecules CrFg , Cr(CN)g , CoFg, and Co(CN)g , for which we have included a selected number of excited states. 

For the Cr and Co compounds, the natural orbital occupation numbers in Table XVI reveal some clear trends, which can be related to the nature of the metal-ligand interaction. First we notice that for the tt donors F , H2O, and NH3, correlation effects within the representa-... 

Natural Orbital Occupation Numbers Resulting from Different CASSCF Calculations on Some Representative Octahedral Transition Metal... 

Table 1 Natural-orbital occupation numbers of the a-spin block of the 1-RDM in the quasi-momentum basis...

Figure 1 Natural orbital occupation numbers for the active orbitals (1-4) in C4H8 as a function of the distance between the two C2H4 fragments. The NOs are shown in Figure 2.

The two natural orbitals a and a will have the occupation numbers p = 2C and p = 2 C, respectively. At infinite distance they will both be one, but near equilibrium almost all of the occupation will reside in the bonding orbital. For weak bonds, an intermediate situation obtains and we can actually define a bond order, BO, from the natural orbital occupation numbers ... 

TABLE lb UHF/MCSCF Natural Orbital occupation Numbers Benzene -Dewar Benzene 6 orbital CAS Transition State Geometry ... 

If the coordinate q is considered to include spin, then the Pauli principle indicates that the natural orbital occupation numbers are between 0 and 1 (inclusive)... 

If spin is not included, then one can put up to two electrons in a single spatial orbital, so that 0 < 2.) An A-electron Slater determinant corresponds to a one-matrix with N unit occupation numbers (corresponding to spin orbitals that appear in the Slater determinant) all the other natural orbital occupation numbers are 0. Because all the eigenvalues of a Slater determinantal one-matrix are 0 or I, the resulting one-matrix is idanpotent ... 

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