Spin-orbitals orthonormalized functions

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. 

The spin functions have the important property that they are orthonormal, i. e., = = 1 and = = 0. For computational convenience, the spin orbitals themselves are usually chosen to be orthonormal also ... 

Now that we have decided on the form of the wave function the next step is to use the variational principle in order to find the best Slater determinant, i. e., that one particular Osd which yields the lowest energy. The only flexibility in a Slater determinant is provided by the spin orbitals. In the Hartree-Fock approach the spin orbitals (Xi 1 are now varied under the constraint that they remain orthonormal such that the energy obtained from the corresponding Slater determinant is minimal... 

Ehf from equation (1-20) is obviously a functional of the spin orbitals, EHF = E[ XJ]. Thus, the variational freedom in this expression is in the choice of the orbitals. In addition, the constraint that the % remain orthonormal must be satisfied throughout the minimization, which introduces the Lagrangian multipliers e in the resulting equations. These equations (1-24) represent the Hartree-Fock equations, which determine the best spin orbitals, i. e., those (xj for which EHF attains its lowest value (for a detailed derivation see Szabo and Ostlund, 1982)... 

In a well known practical but approximate method to solve the GS problem, known as the Hartree-Fock (HF) approximation (see e.g. [10]), the domain of variational functions P in Eq. (9) is narrowed to those that are a single Slater determinant (D) 9 d, constructed out of orthonormal spin orbitals tj/iix) ... 

Here, the j represent the CSFs that are of the correct symmetry, and the Cj are their expansion coefficients to be determined in the variational calculation. If the spin-orbitals used to form the determinants, that in turn form the CSFs j, are orthonormal one-electron functions (i.e., <(). I (f>j> = S j), then the CSFs can be shown to be orthonormal... 

In fact, the Slater determinants themselves also are orthonormal functions of N electrons whenever orthonormal spin-orbitals are used to form the determinants. 

The variational method can be used to optimize the above expectation value expression for the electronic energy (i.e., to make the functional stationary) as a function of the Cl coefficients Cj and the LCAO-MO coefficients Cv, i that characterize the spin-orbitals. However, in doing so the set of Cv,il can not be treated as entirely independent variables. The fact that the spin-orbitals ( )i are assumed to be orthonormal imposes a set of constraints on the Cv,i ... 

The version of HF theory we have been studying is called unrestricted Hartree-Fock (UHF) theory. It is appropriate to all molecules, regardless of the number of electrons and the distribution of electron spins (which specify the electronic state of the molecule). The spin must be taken into account when the exchange integrals are being evaluated since if the two spin orbitals involved in this integral did not have the same spin function, a or ft, the integral value is zero by virtue of the orthonormality of the electron spin functions... 

Consider, for example, two orbitals which might be obtained from a HF calculation on etliylene, namely the orthonormal a and n bonding orbitals, both of which are doubly occupied (Figure D.l). If we resttict our consideration to only these two orbitals, and moreover we use restticted HF theory so that we can ignore the details of spin orbitals, we can write the properly antisymmetric HF wave function for this system of two orbitals as... 

Probably the best-known approach to the utilization of spin symmetry is that originally developed by Slater and by Fock (see, for example, Hurley [17]). No particular advantage is taken of the spin-independence of the Hamiltonian, at least in the first phase of the construction of the n-particle basis. We take the 1-particle basis to be spin-orbitals — products of orthonormal orbitals (r) and the elementary or-thomormal functions of the spin coordinate a... 

The reference state T is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions orbital energy functional E = To + Ec is to be made stationary, subject to the orbital orthonormality constraint (i j) = StJ, imposed by introducing a matrix of Lagrange multipliers. The variational condition is... 

It can be proven [31] that all possible Slater determinants of N particles constructed from a complete system of orthonormalized spin-orbitals 4>k form a complete basis in the space of normalized antisymmetric (satisfying the Pauli principle) functions, of N electrons i.e. for any antisymmetric and normalizable (K one can find expansion amplitudes so that ... 

Restricting the wave function by the form eq. (1.142) allows one to significantly reduce the calculation costs for all characteristics of a many-fermion system. Inserting eq. (1.142) into the energy expression (for the expectation value of the electronic Hamiltonian eq. (1.27)) and applying to it the variational principle with the additional condition of orthonormalization of the system of the occupied spin-orbitals 4>k (known in this context as molecular spin-orbitals) yields the system of integrodiffer-ential equations of the form (see e.g. [27]) ... 

In these methods, the ground state must be of a type that can be described by a single Slater determinant of orthonormal spin orbitals. One such type, which is the most common ground state, is a closed-shell system (i.e. all occupied MOs are doubly occupied). We let or 0 > denote the Slater determinant wave function for the ground state, which will usually be made up of HF MOs, although in fact... 

Assume that you have selected an AO basis set for a given molecular system with N electrons. It has the size m (m molecular orbitals corresponding to 2m spin-orbitals). Transform this basis set in some way to a set of orthonormal one-electron functions. 

The algebraic approximation results in the restriction of the domain of the operator to a finite dimensional subspace, Sf, of the Hilbert space The algebraic approximation may be implemented by defining a suitable orthonormal basis set of M (electron spin orbitals and constructing all unique iV-electron determinants /t> using the M one-electron functions. The... 

We now specialize the discussion to the ligand field theory situation and define the orthonormal set of spin-orbitals we shall use in the determinantal expansion of the many-electron functions Vyy for the groups M and L. First we suppose that we have a set of k orbitals describing the one-electron states in the metal atom these will be orthonormal solutions of a Schrodinger equation for a spherically symmetric potential, V<,(r), which may be thought of as the average potential about the metal atom which an electron experiences ... 

We shall now imagine that the optimized function Vlo for the L group of electrons has been obtained, and look in more detail at the variational calculation specified by Eqs. (3-6 a 3-10) when the functions Wmot are expanded in terms of Slater determinants constructed from the M subset of the orthonormal spin-orbital basis f, the ten spin-orbitals of d-orbital character. 

If the rows of a matrix become linearly dependent, the determinant of that matrix vanishes. This is also true of the determinantal expansion terms. The spin orbitals in the determinant must be linearly independent for the function to have non-zero values for some set of electron coordinates. For an orthonormal spin-orbital basis, this means that no spin orbital may be occupied more than once. The determinantal form of the expansion basis shows that a particular electron cannot be associated with a particular orbital or, accordingly, with a particular region of space. It is still convenient, however, to refer to the electrons in a particular orbital as if the wavefunction were a simple orbital product. The strict interpretation of these references is to the orbital, which is occupied in the determinant by all the electrons, and not to the individual electron which happens to occupy the orbital in some of the Nl terms of the determinant. 

In this subsection, we will briefly discuss how one may construct a basis

carrier space which is adapted not only to the treatment of the ground state of the Hamiltonian H but also to the study of the lowest excited states. In molecular and solid-state theory, it is often natural and convenient to start out from a set of n linearly independent wave functions = < > which are built up from atomic functions (spin orbitals, geminals, etc.) involved and which are hence usually of a nonorthogonal nature due to the overlap of the atomic elements. From this set O, one may then construct an orthonormal set tp = d>A by means of successive, symmetric, or canonical orthonormalization.27 For instance, using the symmetric procedure, one obtains... 

The Hartree-Fock model is the simplest, most basic model in ab initio electronic structure theory [28], In this model, the wave function is approximated by a single Slater determinant constructed from a set of orthonormal spin orbitals ... 

A determinantal wave function expressed in this form is a coherent state. The associated Lie group is the unitary group U K) and the reference state Eq. (127) is the lowest weight state of the irreducible representation [1 0 ] of Lf(K). The stability group is U N) X U(K — N). The norm in an orthonormal basis of spin orbitals is... 

In the Hartree-Fock (HF) model, the n electron wave function. is written as a single Slater determinant of n orthonormal spin orbitals, ( ), ... 

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