Wavefunctions orthogonality

The orbital optimisation is based on the Generalised Brillouin Theorem [8] as extended to non-orthogonal wavefunctions [9,10] ... 

If we have to use non-orthogonal wavefunctions, then the natural one-electron orbitals in which to express them are the SCF molecular orbitals of the non-interacting molecules. From these we can construct antisymmetrized (determinantal) wavefunctions in which some orbitals of each molecule are occupied. Because of the non-orthogonality of the orbitals, these determinant al wavefunctions will also be non-orthogonal. It is possible to construct a perturbation theory in which the wavefunction is expanded in terms of these determinants. Fortunately it is possible to formulate it in such a way that the separation of the Hamiltonian into an unperturbed part and a perturbation is unnecessary. The resulting Intermolecular Perturbation Theory (IMPT) has been incorporated into the Cambridge Analytical Derivatives Package (CADPAC)i ... 

The value of this integral is unity if the wavefunctions are normalized and i and vj/ correspond to the same state. If > and m correspond to different states, the value of the integral will be zero, and the wavefunctions are said to be orthogonal. Wavefunctions that are orthogonal and normalized are called orthonormal. 

Vhen used in this context, the Kronecker delta can be taken to have a value of 1 if m equals n nd zero otherwise. Wavefunctions that are both orthogonal and normalised are said to be rthonormal. 

Tie first consideration is that the total wavefunction and the molecular properties calculated rom it should be the same when a transformed basis set is used. We have already encoun-ered this requirement in our discussion of the transformation of the Roothaan-Hall quations to an orthogonal set. To reiterate suppose a molecular orbital is written as a inear combination of atomic orbitals ... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

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... 

The fii st term is zero because I and its derivatives are orthogonal. The fourth term involves second moments and we use the coupled Hartree-Fock procedure to find the terms requiring the first derivative of the wavefunction. 

The notation is the same as in Exercise 3.45.) Confirm that the bonding and antibonding orbitals are mutually orthogonal— that is, that the integral over the product of the two wavefunctions is zero. 

Because the hydrogen wavefunctions are mutually orthogonal, this sum of integrals simplifies to... 

In the present paper, we propose the use of the HPHF approximation for the direct calculation of excited states, in which M5=0,just as Berthier [11], and Pople and Nesbet [12] did for the determination of states in which Ms 0. We give some examples of such calculations, either when the excited state wavefunction is orthogonal or not by symmetry to that of the ground state. 

A. Express the wavefunction (eigenfunction) as the sum of orthogonal, normalized wavefunctions typically the latter would be spin functions denoted by pj... 

Here are the basic rules of the game For a system with electron spin S, the known complete orthogonal set of 2,S + I wavefunctions is associated with the values ms and is written as... 

If there is a molecular symmetry group whose elements leave the hamiltonian 36 invariant, then the closed-shell wavefunction belongs to the totally symmetric representation of both the spin and symmetry groups.8 It is further true that under these symmetry operations the molecular orbitals transform among each other by means of an orthogonal transformation, such as mentioned in Eq. (5) 9) and, therefore, span a representation of the molecular symmetry group. In general, this representation is reducible. 

At this point it should be noted that, in addition to the discussed previously, the canonical Hartree-Fock equations (26) have additional solutions with higher eigenvalues e . These are called virtual orbitals, because they are unoccupied in the 2iV-electron ground state SCF wavefunction 0. They are orthogonal to the iV-dimensional orbital space associated with this wavefunction. 

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