A, spin function

As is well known, when the electronic spin-orbit interaction is small, the total electronic wave function v / (r, s R) can be written as the product of a spatial wave function R) and a spin function t / (s). For this, we can use either... 

X spin orbital (product of spatial orbital and a spin function)... 

A determinant is the most convenient way to write down the permitted functional forms of a polv electronic wavefunction that satisfies the antisymmetry principle. In general, if we have electrons in spin orbitals Xi,X2, , Xn (where each spin orbital is the product of a spatial function and a spin function) then an acceptable form of the wavefunction is ... 

To avoid having the wave function zero everywhere (an unacceptable solution), the spin orbitals must be fundamentally different from one another. Eor example, they cannot be related by a constant factor. You can write each spin orbital as a product of a space function which depends only on the x, y, and z coordinates of the electron—and a spin function. The space function isusually called the molecular orbital. While an infinite number of space functions are possible, only two spin functions are possible alpha and beta. 

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. 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

Here A is the usual antisymmetrizer (eq. (3.21)) and an bar above an MO indicates that the electron has a (i spin function, no bar indicates an a spin function. 

We may express the single-particle wave function tpniqd fhe product of a spatial wave function 0n(r,) and a spin function % i). For a fermion with spin such as an electron, there are just two spin states, which we designate by a(i) for m = and f i) for Therefore, for two particles there are three... 

The most simple approach is the Hartree-Fock (HF) self-consistent field (SCF) approximation, in which the electronic wave function is expressed as an antisymmetrized product of one-electron functions. In this way, each electron is assumed to move in the average field of all other electrons. The one-electron functions, or spin orbitals, are taken as a product of a spatial function (molecular orbital) and a spin function. Molecular orbitals are constructed as a linear combination of atomic basis functions. The coefficients of this linear combination are obtained by solving iteratively the Roothaan equations. 

We wish to apply permutations, and the antisymmetrizer to products of spin-orbitals that provide a basis for a variational calculation. If each of these represents a pure spin state, the function may be factored into a spatial and a spin part. Therefore, the whole product, 4, may be written as a product of a separate spatial function and a spin function. Each of these is, of course, a product of spatial or spin functions of the individual particles,... 

Here, %i is termed a spin orbital and is the product of a spatial function or molecular orbital, /i, and a spin function, a or 3. ... 

We shall be concerned with ground and excited electronic states which can be adequately described by a single determinantal wave function, i.e. doublet states, triplet states, etc. with spin 5/0). Let 0 be the Slater determinant constructed from a set of spin-orbitals consisting of spatial part ( = 1,2,. ..,n") associated with a spin functions and orbitals... 

Eq.(21) requires that all occupied ground state orbitals be orthogonal to a linear combination of the excited state orbitals b j lvij ), which describes an excited electronic state. Eq.(22) requires the orthogonality of all occupied excited state orbital associated with a spin functions to the arbitrary vector Y7 IVoi ) from the subspace of the occupied ground state orbitals associated with a spin functions. In general, the coefficients 6° can be determined by minimizing the excited state Hartree-Eock energy. However, calculations show that the choice... 

Eq.(29) means that the set of orbitals associated with the (3 spin functions lies completely within the space defined by the set associated with the a spin functions. 

The spatial functions (1.249) and (1.250) must be multiplied by a spin function. For electrons, we have the following four possible two-electron spin functions, three symmetric and one antisymmetric ... 

Since H° is the sum of hydrogenlike Hamiltonians, the zeroth-order wave function is the product of hydrogenlike functions, one for each electron. We call any one-electron spatial wave function an orbital. To allow for electron spin, each spatial orbital is multiplied by a spin function (either a or 0) to give a spin-orbital. To introduce the required antisymmetry into the wave function, we take the zeroth-order wave function as a Slater determinant of spin-orbitals. For example, for the Li ground state, the normalized zeroth-order wave function is... 

In the Hartree-Fock method, the molecular (or atomic) electronic wave function is approximated by an antisymmetrized product (Slater determinant) of spin-orbitals each spin-orbital is the product of a spatial orbital and a spin function (a or ft). Solution of the Hartree-Fock equations (given below) yields the orbitals that minimize the variational integral. Thus the Hartree-Fock wave function is the best possible electronic wave function in which each electron is assigned to a spatial orbital. For a closed-subshell state of an -electron molecule, minimization... 

Such an approach leads to a total wave function for the system, which is an anti-symmetrized product of molecular spin orbitals (spin orbital = molecular orbital times a spin function). The Hartree-Fock method is obtained by applying the variation principle to the corresponding energy functional. 

The Hartree wavefunction (above) is a product of one-electron functions called orbitals, or, more precisely, spatial orbitals these are functions of the usual space coordinates x, y, z. The Slater wavefunction is composed, not just of spatial orbitals, but of spin orbitals. A spin orbital ij/ (spin) is the product of a spatial orbital and a spin function, a or / The spin orbitals corresponding to a given spatial orbital are... 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

The 16 spin orbitals in this determinant are the Kohn-Sham spin orbitals of the reference system each is the product of a Kohn-Sham spatial orbital y/i and a spin function a or jS. Equation 7.18 can be written in terms of the spatial KS orbitals by invoking a set of rules (the Slater-Condon rules [34]) for simplifying integrals involving Slater determinants ... 

In the orbital description of electronic structure, the motion of each electron is determined by a spin-orbital, ut, a simple product of a space function fc and a spin function at... 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

The N-electron wave-function of a polyatomic molecule containing the atoms A, B. n is usually most conveniently expressed as a Slater product of molecular spin orbitals each of which is a linear combination of atomic orbitals multiplied by a spin function... 

There are three restrictions that are normally incorporated into Hartree-Fock calculations, and a fourth often appears when the Hartree-Fock formalism is used to parametrize the experimental results. (1) The spacial part of a one-electron wave function pi is assumed to be separable into a radial and an angular part, so that = r lUi(r)Si(e,)Si(a) where Si(a) is a spin function with spin... 

Hartree-Fock-Roothaan Closed-Shell Theory. Here [7], the molecular spin-orbitals it where the subscript labels the different MOs, are functions of (af, 2/", z") (where /z stands for the coordinate of the /zth electron) and a spin function. The configurational wave function is represented by a single determinantal antisymmetrized product wave function. The total Hamiltonian operator 2/F is defined by... 

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