Spin orbitals orthonormal

Here, the d>j represent the CSFs that are of the eorreet symmetry, and the Cj are their expansion eoeffieients to be determined in the variational ealeulation. If the spin-orbitals used to form the determinants, that in turn form the CSFs Oj, are orthonormal one-eleetron funetions (i.e., <(l)k (l)j> = 5kj), then the CSFs ean be shown to be orthonormal funetions of N eleetrons... 

In faet, the Slater determinants themselves also are orthonormal funetions of N eleetrons whenever orthonormal spin-orbitals are used to form the determinants. 

The variational method ean be used to optimize the above expeetation value expression for the eleetronie energy (i.e., to make the funetional stationary) as a funetion of the Cl eoeffieients Cj and the ECAO-MO eoeffieients Cv,i that eharaeterize the spin-orbitals. However, in doing so the set of Cv,i ean not be treated as entirely independent variables. The faet that the spin-orbitals ([ti are assumed to be orthonormal imposes a set of eonstraints on the Cv,i ... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

In eontrast, if Pi and T 2 are obtained by earrying out a Cl ealeulation using a single set of orthonormal spin-orbitals (e.g., with Pi and P2 formed from two different... 

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. 

Molecular rearrangement resulting from molecular collisions or excitation by light can be described with time-dependent many-electron density operators. The initial density operator can be constructed from the collection of initially (or asymptotically) accessible electronic states, with populations wj. In many cases these states can be chosen as single Slater determinants formed from a set of orthonormal molecular spin orbitals (MSOs) im as / =... 

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)... 

The HF spin orbitals %, that appear in this expression are chosen such that the expectation value Ehf attains its minimum (under the usual constraint that the remain orthonormal)... 

The energy expression for a single determinant 4>0 of orthonormal spin-orbitals i>i is... 

In what follows the number N of electrons of the system under study is a fixed number, and the dimension of the finite Hilbert subspace spanned by the orthonormal basis of spin-orbitals is 2K. 

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) ... 

In what follows, the number N of particles is assumed to be a constant. The one-electron basis is assumed to be finite and formed by 2JC orthonormal spin orbitals denoted by the italic letters i,j, k,l... or, when the spin is considered explicitly by ia or. ... 

In what follows a two-particle interacting system having a hxed and weU-dehned number of particles N will be considered. It will also be considered that the one-electron space is spanned by a hnite basis set of 2K orthonormal spin orbitals. Under these conditions the 1-RDM and 2-RDM elements are dehned in second... 

Most of this chapter utilizes the first-quantized formulation of the ROMs introduced above. However, some concepts related to separabihty and extensiv-ity are more easily discussed in second quantization, and the second-quantized formalism is therefore employed in Section IE. Introducing an orthonormal spin-orbital basis 1 ) = dj 0), the elements of the p-RDM are expressed directly in second quantization as... 

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

In our formalism [5-9] excitation operators play a central role. Let an orthonormal basis p of spin orbitals be given. This basis has usually a finite dimension d, but it should be chosen such that in the limit —> cxd it becomes complete (in the so-called first Sobolev space [10]). We start from creation and annihilation operators for the ij/p in the usual way, but we use a tensor notation, in which subscripts refer to annihilation and superscripts to creation ... 

The condition that the SCF energy be stationary with respect to variations 8( )j in the occupied spin-orbitals (that preserve orthonormality) can be written... 

Please minimize the electronic energy of this single determinant subject to the constraint that the spin orbitals all remain orthonormal to one another. 

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... 

Let 0i(r,a), i = 1, m denote a basis of m orthonormal spin orbitals where r are the spatial coordinates and o is the spin coordinate. A Slater determinant is an antisymmetric linear combination of one or more of these spin orbitals. The occupation of a given Slater determinant can be written as an occupation number vector, ln>, where nj is one if spin orbital j is occupied in the Slater determinant and nj is zero if spin orbital (j> is unoccupied. 

For an orthonormal basis of spin orbitals, we define the inner product between two occupation number vectors I n> and I k> as... 

The discussion in the previous sections dealt with the second quantization formalism for orthonormal spinorbitals. In this section, we will generalize the formalism to treat cases where the set of spin orbitals is non-orthogonal. Consider a set of n spin orbitals with the general metric... 

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... 

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