Doubly-occupied

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). 

Spin orbitals arc grouped in pairs for an KHF ealetilation, Haeti mem her of ih e pair dilTcrs in its spin function (one alpha and one beta), hilt both must share the same space function. For X electrons, X/2 different in olecu lar orbitals (space function s larc doubly occupied, with one alpha (spin up) and one beta (spin down) electron forming a pair. 

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

After diagonalization of the EHT matrix, the lowest 4 orbitals have an energy sum of about —70 eV. The electronic energy for these doubly occupied orbitals is 2(—70) = — 140 eV. The energy gain of the molecule relative to its atoms is —140 — ( — 110) = —30eV = —690 kcal mol (1 eV = 23 kcal mol ) therefore, the molecule is stable relative to its atoms. We can envision an energy cycle with three steps (Eig. 7-5) ... 

The total it electron energy is the sum of occupied orbital energies multiplied by two if. as is usually the ease, the orbital is doubly occupied. The charge densities and free valency indices were treated in separate sections above. The bond order output should be read as a lower triangular serni matrix. The bond order semi matrix for the butadiene output is shown in Fig. 7-7. 

The eigenstates /n(x) and their energies En represent orbitals into whieh eleetrons are plaeed. In the example ease, if nine n eleetrons are present (e.g., as in the 1,3,5,7-nonatetraene radieal), the ground eleetronie state would be represented by a total wavefunetion eonsisting of a product in which the lowest four / s are doubly occupied and the fifth / is singly occupied ... 

An example will help illustrate these ideas. Consider the formaldehyde molecule H2CO in C2v symmetry. The configuration which dominates the ground-state waveflinction has doubly occupied O and C 1 s orbitals, two CH bonds, a CO a bond, a CO n bond, and two 0-centered lone pairs this configuration is described in terms of symmetry adapted orbitals as follows (Iai22ai23ai2lb2 ... 

In the specific case considered here, X(E) = 4, X(2C3) = 1, and X(3av) = 0 (You should try this.). Notice that the contributions of any doubly occupied nondegenerate orbitals... 

If IS offen convenienf to speak of the valence electrons of an atom These are the outermost electrons the ones most likely to be involved m chemical bonding and reac tions For second row elements these are the 2s and 2p electrons Because four orbitals (2s 2p 2py 2pf) are involved the maximum number of electrons m the valence shell of any second row element is 8 Neon with all its 2s and 2p orbitals doubly occupied has eight valence electrons and completes the second row of the periodic table... 

Bonding orbital (Section 2 4) An orbital in a molecule in which an electron is more stable than when localized on an isolated atom All the bonding orbitals are normally doubly occupied in stable neutral molecules... 

Hund s rule (Section 1 1) When two orbitals are of equal en ergy they are populated by electrons so that each is half filled before either one is doubly occupied Hybrid orbital (Section 2 6) An atomic orbital represented as a mixture of vanous contributions of that atom ss p d etc orbitals... 

If a covalent bond is broken, as in the simple case of dissociation of the hydrogen molecule into atoms, then theRHFwave function without the Configuration Interaction option (see Extending the Wave Function Calculation on page 37) is inappropriate. This is because the doubly occupied RHFmolecular orbital includes spurious terms that place both electrons on the same hydrogen atom, even when they are separated by an infinite distance. 

Asimple example is the formation of the hydrogen molecule from two hydrogen atoms. Here the original atomic energy levels are degenerate (they have equal energy), but as the two atoms approach each other, they interact to form two non degenerate molecular orbitals, the lowest of which is doubly occupied. 

Although not strictly part of a model chemistry, there is a third component to every Gaussian calculation involving how electron spin is handled whether it is performed using an open shell model or a closed shell model the two options are also referred to as unrestricted and restricted calculations, respectively. For closed shell molecules, having an even number of electrons divided into pairs of opposite spin, a spin restricted model is the default. In other words, closed shell calculations use doubly occupied orbitals, each containing two electrons of opposite spin. 

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)). 

The simplest antisymmetric function that is a combination of molecular orbitals is a determinant. Before forming it, however, we need to account for a factor we ve neglected so far electron spin. Electrons can have spin up i+Vi) or down (-V2). Equation 20 assumes that each molecular orbital holds only one electron. However, most calculations are closed shell calculations, using doubly occupied orbitals, holding two electrons of opposite spin. For the moment, we will limit our discussion to this case. 

There are m doubly occupied molecular orbitals, and the number of electrons is 2m because we have allocated an a and a spin electron to each. In the original Hartree model, the many-electron wavefunction was written as a straightforward product of one-electron orbitals i/p, i/ and so on... 

Here, c is a column vector of LCAO coefficients and e is called the orbital energy. If we start with n basis functions, then there are exactly n different c s (and e s) and the m lowest-energy solutions of the eigenvalue problem correspond to the doubly occupied HF orbitals. The remaining n — m solutions are called the virtual orbitals. They are unoccupied. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

You probably noted that the original papers were couched in terms of HF-LCAO theory. From Chapter 6, the defining equation for a Hamiltonian matrix element (in the usual doubly occupied molecular orbital, closed-shell case) is... 

In Chapter 6, I discussed the open-shell HF-LCAO model. 1 considered the simple case where we had ti doubly occupied orbitals and 2 orbitals all singly occupied by parallel spin electrons. The ground-state wavefunction was a single Slater determinant. I explained that it was possible to derive an expression for the electronic energy... 

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