Molecular orbital defined

Since the operators f and do not commute, the order of the two operations is not irrelevant. The order used above is the one commonly used in the formulation of the problem. The reverse order can be shown to lead to much more complex algebra, needed to determine the transformed state Owing to the definitions of and f, the varied wavefunction (27) will remain normalized and the new molecular orbitals defined as... 

We now consider the CSF contribution to the first derivative nonadiabatic coupling matrix element. A CSF, 06, is an antisymmetric, symmetry-adapted product of molecular orbitals, defined in Eq. (5b) so that its derivative has the form... 

Group theory can be used to resolve this distinction, because the sets of paired molecular orbitals defined above are related by specific symmetry relationships. If ] ... 

If you define a density matrix R by summing over all occupied molecular orbitals ... 

The origins of the Finnis-Sinclair potential [Finnis and Sinclair 1984] lie in the density of states and the moments theorem. Recall that the density of states D(E) (see Section 3.8.5) describes the distribution of electronic states in the system. D(E) gives the number of states between E and E - - 8E. Such a distribution can be described in terms of its moments. The moments are usually defined relative to the energy of the atomic orbital from which the molecular orbitals are formed. The mth moment, fi", is given by ... 

The Jacobi method is probably the simplest diagonalization method that is well adapted to computers. It is limited to real symmetric matrices, but that is the only kind we will get by the formula for generating simple Huckel molecular orbital method (HMO) matrices just described. A rotation matrix is defined, for example. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

The Exclusion Principle is fundamentally important in the theory of electronic structure it leads to the picture of electrons occupying distinct molecular orbitals. Molecular orbitals have well-defined energies and their shapes determine the bonding pattern of molecules. Without the Exclusion Principle, all electrons could occupy the same orbital. 

Energy, geometry, dipole moment, and the electrostatic potential all have a clear relation to experimental values. Calculated atomic charges are a different matter. There are various ways to define atomic charges. HyperChem uses Mulliken atomic charges, which are commonly used in Molecular Orbital theory. These quantities have only an approximate relation to experiment their values are sensitive to the basis set and to the method of calculation. 

The Roothaan equations are the basic equations for closed-shell RHF molecular orbitals, and the Pople-Nesbet equations are the basic equations for open-shell UHF molecular orbitals. The Pople-Nesbet equations are essentially just the generalization of the Roothaan equations to the case where the spatials /j and /jP, as shown previously, are not defined to be identical but are solved independently. 

The unrestricted approach defines two different sets of spatial molecular orbitals—those that hold electrons with spin up ... 

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. 

Multiplying a molecular orbital function by a or P will include electron spin as part of the overall electronic wavefunction i /. The product of the molecular orbital and a spin function is defined as a spin orbital, a function of both the electron s location and its spin. Note that these spin orbitals are also orthonormal when the component molecular orbitals are. 

We can now build a closed shell wavefunction by defining n/2 molecular orbitals for a system with n electrons, and then assigning electrons to these orbitals in pairs of opposite spin ... 

A convenient orbital method for describing eleetron motion in moleeules is the method of molecular orbitals. Molecular orbitals are defined and calculated in the same way as atomic orbitals and they display similar wave-like properties. The main difference between molecular and atomic orbitals is that molecular orbitals are not confined to a single atom. The crests and troughs in an atomic orbital are confined to a region close to the atomic nucleus (typieally within 1-2 A). The electrons in a molecule, on the other hand, do not stick to a single atom, and are free to move all around the molecule. Consequendy, the crests and troughs in a molecular orbital are usually spread over several atoms. 

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... 

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