Optimization, orbital

Where X is an antisymmetric matrix containing the independent (orthogonal) rotation parameters. Expanding the energy in X about the origin 

The one- and two-body density matrices are formed by contraction of the vector coupling coefficients with the left and right eigenvectors of Eq. (10) (assuming CSC = 1) 

Further details of the evaluation of Ro, such as approximate expressions for the hessian matrix, G, in Eq. (20), may be found in Ref. 10. 

This produces a non-standard Rumer function as shown by the representation on the right hand side. These non-standard functions, which correspond to a Rumer diagram in which the lines cross, are not used in calculations. A standard set of functions can be selected using the following procedure the open shell orbitals are represented by a 1 if their number occurs in the left hand column of the Weyl tableau or a 2 if it appears in the right hand column. Hence for the example above 

We now pair the left most 2 with the closest 1 to its left, i.e. 

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Chaban G, Schmidt M W and Gordon M S 1997 Approximate second order methods for orbital optimization of SCF and MCSCF wavefunctlons Theor. Chim. Acta 97 88... 

For sueh a funetion, the CI part of the energy minimization is absent (the elassie papers in whieh the SCF equations for elosed- and open-shell systems are treated are C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951) 32, 179 (I960)) and the density matriees simplify greatly beeause only one spin-orbital oeeupaney is operative. In this ease, the orbital optimization eonditions reduee to ... 

Before addressing head-on the problem of how to best treat orbital optimization for open-shell species, it is useful to examine how the HF equations are solved in practice in terms of the LCAO-MO process. 

All of the CSFs in the SCF (in which case only a single CSF is included) or MCSCF wavefunction that was used to generate the molecular orbitals (jii. This set of CSFs are referred to as spanning the reference space of the subsequent CI calculation, and the particular combination of these CSFs used in this orbital optimization (i.e., the SCF or MCSCF wavefunction) is called the reference function. 

Another technique, called Brueckner doubles, uses orbitals optimized to make single excitation contributions zero and then includes double excitations. This is essentially equivalent to CCSD in terms of both accuracy and CPU time. 

The simplest description of an excited state is the orbital picture where one electron has been moved from an occupied to an unoccupied orbital, i.e. an S-type determinant as illustrated in Figure 4.1. The lowest level of theory for a qualitative description of excited states is therefore a Cl including only the singly excited determinants, denoted CIS. CIS gives wave functions of roughly HF quality for excited states, since no orbital optimization is involved. For valence excited states, for example those arising from excitations between rr-orbitals in an unsaturated system, this may be a reasonable description. There are, however, normally also quite low-lying states which essentially correspond to a double excitation, and those require at least inclusion of the doubles as well, i.e. CISD. 

THEORY OF ORBITAL OPTIMIZATION IN SCT AND MCSF CALCULATIONS form of a power expansion... 

TEEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS with... 

Because Rydberg states are peculiar states with a core resembling the positive ion and one electron in a diffuse orbital, the A] Rydberg states have been recalculated with orbitals optimized for the ion, with the same MCSCF/SD expansion. An improvement of 0.2 eV is obtained, arguing for the use of this type of MOs for Rydberg A, states. 

The same conclusion, that MCSCF/SD expansions using orbitals optimized for the ion provide a better representation, is reached for the lowest states of 82 symmetry which are also states of Rydberg type arising from an in-plane excitation from the carbene orbital. 

No significant improvement for the vertical excitation energy of the 2 B (3p) state was found. From these results we have decided to describe the lowest states of B and A2 symmetries with the same set of molecular orbitals, optimized for the neutral molecule within the MCSCF/ 6422 expansion. 

Our best estimation for the vertical excitation energies for states of A, symmetry are reported in Table 12. They correspond to a ground state calculated at CI( 6) level using orbitals optimized for the neutral molecule with the MCSCF/SD expansion, and excited Rydberg states calculated at the level using orbitals optimized for the positive ion... 

Vertical excitation energies to states of B symmetry, calculated at the level using the orbitals optimized for the neutral molecule with the MCSCF/6422 expansion, are reported Table 12. The I Bi valence state and 2 B (3p) Rydberg state of C3H2 are respectively 5.2 eV and 7.5 eV above the ground state with large transition moments of... 

Finaly, the lowest two Aj states, calculated at the CI(,.6)level using orbitals optimized for the neutral molecule with the MCSCF/ 6422 expansion are at 4.66 eV and 8.59 eV. Transitions from the Aj to these states are not symmetry-allowed and it is hardly probable that vibronic coupling could make them observable in transient conditions. 

In this model, one considers the acetals to be composed of polarizable dipolar moieties that can be stabilized by electron transfer from an electron-rich moiety (non-bonding electron on oxygen low ionisation energy) to adjacent polar and polarizable moieties (high electron affinity). A strong overlap between n(O) and cr c 0 orbitals optimizes this electronic transfer. As these orbitals are not spherical, n(0)/ct c o overlap depends on... 

The final step is the orbital optimization for the truncated SDTQ-CI expansion. We used the Jacobi-rotation-based MCSCF method of Ivanic and Ruedenberg (55) for that purpose. Table 1 contains the results for the FORS 1 and FORS2 wavefiinctions of HNO and NCCN, obtained using cc-pVTZ basis sets (51). In all cases, the configurations were based on split-localized orbitals. For each case, four energies are listed corresponding to (i) whether the full or the truncated SDTQ-CI expansion was used and (ii) whether the split-localized orbitals were those deduced from the SD naturals orbitals or were eventually MCSCF optimized. It is seen that... 

Table 1. Energies of full and truncated SDTQ-Cl FORS-1 and FORS-2 calculations with and without MCSCF orbital optimization for the molecules HNO and NCCN (values in Hartrees unless noted otherwise).

As is the case for standard orthogonal-orbital MCSCF calculations, the optimization of VB wavefimctions can be a complicated task, and a program such as CASVB should therefore not be treated as a black box . This is true, to a greater or lesser extent, for most procedures that involve orbital optimization (and, hence, non-linear optimization problems), but these difficulties are compounded in valence bond theory by the... 

Intra-Orbit Optimization of Energy Density Functionals. 205... 

Euler-Lagrange Equation for Intra-Orbit Optimization of p(r) 206... 

Euler-Lagrange Equations for the Intra-Orbit Optimization of... 

In Fig. 7, we present a general scheme, comprising the intra- and inter-orbit optimizations appearing in the variational problem described by Eq. (138). We discuss this optimization process with reference to only three of the infinite number of orbits into which Hilbert space is decomposed. These orbits are... 

Fig. 7. Schematic representation of intra-orbit and inter-orbit optimizations...

The intra-orbit optimization process is shown schematically in each one of the columns of Fig. 7, underneath the orbit symbol. Thus, with particular reference to the column below (P, let us assume that the orbit-generating function is chosen. Since this function is explicitly known, we can obtain explicit expres-... 

As indicated in Fig. 7, the next step after either an explicit or an implicit energy density functional auxiliary functional Q[p(r) made up of the energy functional [p(r) 9 ]. plus the auxiliary conditions which must be imposed on the variational magnitudes. Notice that there are many ways of carrying out this variation, but that - in general - one obtains Euler-Lagrange equations by setting W[p(r) = 0. 

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