Virtual molecular orbitals

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Even with the minimal basis set of atomic orbitals used m most sem i-empirical calculatitm s. the n urn ber of molecii lar orbitals resulting from an SCFcalciilation exceeds the num ber of occupied molecular orbitals by a factor of about two. The n um ber of virtual orbitals in an ah initio calculation depends on the basis set used in this calculation. 

For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

A Hbasis functions provides K molecular orbitals, but lUJiW of these will not be occupied by smy electrons they are the virtual spin orbitals. If u c were to add an electron to one of these virtual orbitals then this should provide a means of calculating the electron affinity of the system. Electron affinities predicted by Konpman s theorem are always positive when Hartree-Fock calculations are used, because fhe irtucil orbitals always have a positive energy. However, it is observed experimentally that many neutral molecules will accept an electron to form a stable anion and so have negative electron affinities. This can be understood if one realises that electron correlation uDiild be expected to add to the error due to the frozen orbital approximation, rather ihan to counteract it as for ionisation potentials. 

The Bom-Oppenheimer approximation is not peculiar to the Huckel molecular orbital method. It is used in virtually all molecular orbital calculations and most atomic energy calculations. It is an excellent approximation in the sense that the approximated energies are very close to the energies we get in test cases on simple systems where the approximation is not made. 

Pop=Reg Displays highest five occupied and lowest five virtual molecular orbitals and other information not included in the output by default. Use PopsFull to display all orbitals. 

The atomic orbital contributions for each atom in the molecule are given for each molecular orbital, numbered in order of increasing energy (the MO s energy is given in the row labeled EIGENVALUES preceding the orbital coefficients). The symmetry of the orbital and whether it is an occupied orbital or a virtual (unoccupied) orbital appears immediately under the orbital number. 

The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) may be identified by finding the point where the occupied/virtual code letter in the symmetry designation changes from O to V. 

Include the keywords IOP(9/40=3) and Pop=Full in the route section of your jobs. The latter requests that all molecular orbitals (occupied and virtual) be included in the population analysis, while the former specifies that all wavefunctioii coefficients greater than 0.001 be included in the excited state output (by default, only those greater than 0.1 are listed). 

A CASSCF calculation is requested in Gaussian with the CASSCF keyword, which requires two integer arguments the number of electrons and the number of orbitals in the active space. The active space is defined assuming that the electrons come from as many of the highest occupied molecular orbitals as are needed to obtain the specified number of electrons any remaining required orbitals are taken from the lowest virtual orbitals. 

For example, in a 4-electron, 6-orbital CAS—specified as CASSCF 4,6)—performed on a singlet system, the active space would consist of the two highest occupied molecular orbitals (where the four electrons reside) and the four lowest virtual orbitals. Similarly, for a 6-electron, 5-orbital CAS on a triplet system, the active space would consist of the four highest occupied MO s— two of which are doubly-occupied and two are singly-occupied—and the LUMO (the keyword is CASSCF(6,5)). 

Just as the variational condition for an HF wave function can be formulated either as a matrix equation or in terms of orbital rotations (Sections 3.5 and 3.6), the CPFIF may also be viewed as a rotation of the molecular orbitals. In the absence of a perturbation the molecular orbitals make the energy stationary, i.e. the derivatives of the energy with respect to a change in the MOs are zero. This is equivalent to the statement that the off-diagonal elements of the Fock matrix between the occupied and virtual MOs are zero. 

The CPHF equations are linear and can be determined by standard matrix operations. The size of the U matrix is the number of occupied orbitals times the number of virtual orbitals, which in general is quite large, and the CPHF equations are normally solved by iterative methods. Furthermore, as illustrated above, the CPHF equations may be formulated either in an atomic orbital or molecular orbital basis. Although the latter has computational advantages in certain cases, the former is more suitable for use in connection with direct methods (where the atomic integrals are calculated as required), as discussed in Section 3.8.5. 

When enantiomerically pure allyl p-tolyl sulfoxide is deprotonated and then treated with electrophilic 2-cyclopentenone, a conjugate addition occurs forming a new carbon-carbon bond with very high control of absolute stereochemistry (equation 25)65. See also Reference 48. Similarly, using more substituted enantiomerically pure allylic sulfoxides leads to virtually complete diastereocontrol, as exemplified by equations 26 and 27 the double bond geometry in the initial allylic sulfoxide governs the stereochemistry at the newly allylic carbon atom (compare equations 26 vs. 27)66. Haynes and associates67 rationalize this stereochemical result in terms of frontier molecular orbital considerations... 

Figure 3. Molecular-orbital diagrams as obtained by the ROHF method. Dashed lines indicate MOs dominated by the metal d-orbitals, the solid lines stand for doubly occupied or virtual ligand orbitals. Orbitals which are close in energy are presented as degenerate the average deviation from degeneracy is approximately 0.01 a.u. In the case of a septet state (S=3), the singly occupied open-shell orbitals come from a separate Fock operator and their orbital energies do not relate to ionization potentials as do the doubly occupied MOs (i.e. Koopmann s approximation). For these reasons, the open-shell orbitals appear well below the doubly occupied metal orbitals. Doubly occupying these gives rise to excited states, see text.

We will now discuss an iterative scheme based on the CHF approach outlined in Sections 11 and 111, using the McWeeny procedure [7] for resolving matrices into components, by introducing projection operators R and R2, with respect to the subspaces spanned by occupied and virtual molecular orbitals. 

All three states were described by a single set of SCF molecular orbitals based on the occupied canonical orbitals of the X Z- state and a transformation of the canonical virtual space known as "K-orbitals" [10] which, among other properties, approximate the set of natural orbitals. Transition moments within orthogonal basis functions are easier to derive. For the X state the composition of the reference space was obtained by performing two Hartree-Fock single and double excitations (HFSD-CI) calculations at two typical intemuclear distances, i.e. R. (equilibrium geometry) and about 3Re,and adding to the HF... 

Use in first order of the Cl coefficients to correct the occupied and virtual molecular orbitals... 

The naphthalene anion radical spectrum (Figure 2.2) provided several surprises when Samuel Weissman and his associates1 first obtained it in the early 1950s at Washington University in St. Louis. It was a surprise that such an odd-electron species would be stable, but in the absence of air or other oxidants, [CioHg]- is stable virtually indefinitely. A second surprise was the appearance of hyperfine coupling to the two sets of four equivalent protons. The odd electron was presumed (correctly) to occupy a it molecular orbital... 

The quality of the TD-DFT results is determined by the quality of the KS molecular orbitals and the orbital energies for the occupied and virtual states. These in turn depend on the exchange-correlation potential. In particular, excitations to Rydberg and valence states are sensitive to the behavior of the exchange-correlation potential in the asymptotic region. If the exchange-correla-... 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

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