Closed-shells restricted Hartree-Fock

You can order the molecular orbitals that are a solution to equation (47) according to their energy. Electrons populate the orbitals, with the lowest energy orbitals first. Anormal, closed-shell, Restricted Hartree Fock (RHF) description has a maximum of two electrons in each molecular orbital, one with electron spin up and one with electron spin down, as shown ... 

The Roothaan equations just described are strictly the equations for a closed-shell Restricted Hartree-Fock (RHF) description only, as illustrated by the orbital energy level diagram shown earlier. To be more specific ... 

The formal justification of this expansion form is analogous to that given by Hurley, Lennard-Jones and Pople in the development of a correlated-pair extension of the closed-shell restricted Hartree-Fock (RHF) wavefunction. 

Figwe2.10 A closed-shell restricted Hartree-Fock ground state determinant,... 

Having described restricted closed-shell Hartree-Fock calculations in conjunction with the H2 and HeH model systems, we now want to illustrate the results of more realistic calculations on polyatomic molecules. We do this not to provide a review of current calculations, but rather to illustrate the main ideas behind all calculations of the closed-shell restricted Hartree-Fock type, and to provide some feeling and intuition for how well (or how... 

While most of the focus on and use of localized orbitals has been with regard to closed shell restricted Hartree-Fock (RHF) wavefunctions. the applications of localization criteria are considerably more general. Of particular importance is the fact that complex multi-conligurational wavefunctions can also be localized, as long as the full active space of electrons and orbitals, according to the FORS or CASSCF prescription, is included. MCSCF localized orbitals are particularly useful for the interpretation of bonding. Consider, for example, a typical double bond between a transition metal (M) and a main group element (E). A simple RHF description of such a bond is provided by the wavefunction... 

Briefly, the starting point for M ller-Plesset calculations is either an unrestricted Hartree-Fock (UHF) calculation or a closed-shell restricted Hartree-Fock (RHF) calculation. The zero order Hamiltonian, Hq, is then taken to be the sum of the one-electron Fock operators, Fp, and the perturbation, V, is defined as the difference between Hq and the full, many-electron Hamiltonian, ducing an expansion parameter, X, then gives. 

Yamaguchi et al. [182] have applied the equations derived by diem for analytic simultaneous evaluation of vibrational frequencies and intensities for closed-shell, open-shell imrestricted and open-shell restricted Hartree-Fock wave functions for a number of molecules using basis sets of different complexity. Their results illustrate very clearly the basis set dependence of calculated vibrational parameters. Some of their results are represented in Table 7.2. Several molecular quantities are included since the comparisons for the accuracy of predicted values are quite interesting. Even larger basis have been used by Amos in consistent analytic derivative calculations of harmonic frequencies, infrared and Raman intensities for H2O, NH3 and CH4 as test molecules [175,189]. The results for H2O, HF, CO, NH3, CH4 and C2H2 of Yamaguchi et al. [182] and Amos [175, 189] are summarized in Tables 7.3 and 7.4. 

Choose UHF (spin Unrestricted Hartree-Fock) or RHF (spin Restricted Hartree-Fock) calculations according to your molecular system. HyperChem supports UHF for both open-shell and closed-shell calculations and RHF for closed-shell calculations only. The closed-shell UHF calculation may be useful for studying dissociation of molecular systems. ROHF (spin Restricted Open-shell Hartree-Fock) is not supported in the current version of HyperChem (for ab initio calculations). 

Here we give the molecule specification in Cartesian coordinates. The route section specifies a single point energy calculation at the Hartree-Fock level, using the 6-31G(d) basis set. We ve specified a restricted Hartree-Fock calculation (via the R prepended to the HF procedure keyword) because this is a closed shell system. We ve also requested that information about the molecular orbitals be included in the output with Pop=Reg. 

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. 

At this point, it is appropriate to draw a parallel with the straightforward MO explanations for the aromaticity of benzene using approaches based on a single closed-shell Slater determinant, such as HMO and restricted Hartree-Fock (RWF), which also have no equivalent within more advanced multi-configuration MO constructions. The relevance of this comparison follows from the fact that aromaticity is a primary factor in at least one of the popular treatments of pericyclic reactions Within the Dewar-Zimmerman approach [4-6], allowed reactions are shown to pass through aromatic transition structures, and forbidden reactions have to overcome high-energy antiaromatic transition structures. 

The method of calculating wavefunctions and energies that has been described in this chapter applies to closed-shell, ground-state molecules. The Slater determinant we started with (Eq. 5.12) applies to molecules in which the electrons are fed pairwise into the MO s, starting with the lowest-energy MO this is in contrast to free radicals, which have one or more unpaired electrons, or to electronically excited molecules, in which an electron has been promoted to a higher-level MO (e.g. Fig. 5.9, neutral triplet). The Hartree-Fock method outlined here is based on closed-shell Slater determinants and is called the restricted Hartree-Fock method or RHF method restricted means that the electrons of a spin are forced to occupy (restricted to) the same spatial orbitals as those of jl spin inspection of Eq. 5.12 shows that we do not have a set of a spatial orbitals and a set of [l spatial orbitals. If unqualified, a Hartree-Fock (i.e. an SCF) calculation means an RHF calculation. 

Pople refers to a specific set of approximations as defining a theoretical model. Hence the ab initio or Hartree-Fock models employ the Born-Oppenheimer, LCAO and SCF approximations. If the system under study is a closed-shell system (even number of electrons, singlet state), the constraint that each spatial orbital should contain two electrons, one with a and one with P spin, is normally made. Such wavefunctions are known as restricted Hartree-Fock (RHF). Open-shell systems are better described by unrestricted Hartree-Fock (UHF) wavefunctions, where a and P electrons occupy different spatial orbitals. We have seen that Hartree-Fock (HF) models give rather unreliable energies. 

A restricted Hartree-Fock calculation on a closed-shell n-electron system using a basis set of N orbitals will produce n/2 doubly-occupied molecular orbitals and N—nj2 vacant or virtual orbitals. In a standard Cl calculation, the excited-state determinants are formed by systematically promoting electrons from the occupied orbitals of the ground-state determinant to the vacant or virtual orbitals. The number of configurations which can be formed in this way from N electrons and n basis functions178 is of the order of nN. Thus, even with today s high speed computers, a full Cl is possible only for very small systems. 

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 many-electron wave function of a molecular system is taken as the antisymmetrized product of (pt, and for closed-shell systems it is convenient to represent it by a Slater determinant. Such an approach is known as the restricted Hartree-Fock (RHF) method and is the most widely used method in chemisorption calculations. Its principal drawback is the neglect of Coulomb electron correlation, which is of crucial importance for adequate treatment of chemical rearrangements with varying numbers of electron pairs. 

Bubble Diagrams.—The bubble diagrams which are shown in Figures 13 and 14 are required when the reference function for a closed-shell system is not defined by the matrix Hartree-Fock model, when a restricted Hartree-Fock reference... 

In the later part of the 1950 s, It was evident that it was necessary to distlngush the new approach dealing with different orbitals for a-spln and P-spln from the previous approach starting out from symmetry restrictions the latter was called the Restricted Hartree-Fock (RHF) scheme, whereas the new approach was called the Unrestricted Hartree-Fock (UHF) scheme. For some time there was a certain amount of competition between the two schemes. In the late 1950 s, it was further shown that the RHF-scheme for closed-shell systems was completely se[f-consistent not only for atoms but also for molecules and solids [16.17] and that, if one started by imposing a symmetry requirement on the original Slater determinant, this assumption would be self-consistent, i.e. the final determinant would have the same symmetry property. Since symmetry properties are of such fundamental importance in quantum theory, one would hence anticipate that the RHF-scheme would... 

The preceding development of the HF theory assumed a closed-shell wavefunction. The wavefunction for an individual electron describes its spatial extent along with its spin. The electron can be either spin up (a) or spin down (P). For the closed-shell wavefunction, each pair of electrons shares the same spatial orbital but each has a different spin—one is up and the other is down. This type of wavefunction is also called a (spin)-restricted wavefunction since the paired electrons are restricted to the same spatial orbital, leading to the restricted Hartree-Fock (RHF) method. 

There are a number of slightly more approximate methods for determining the electron affinity (EA) based on the restricted Hartree-Fock (RHF) and spin-unrestricted Hartree-Fock (UHF) methods. For closed shell anions, molecules which dissociate to... 

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 j3 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 Rp.strirted Open-shell Hartree-Fock (RQHF). For open-shell species a UHF treatment... 

Closed-shell (diamagnetic) systems can be investigated using a restricted Hartree-Fock (RHF) calculation, while unrestricted Hartree-Fock (UHF) calculations are able to accommodate open-shell (paramagnetic) systems as well. The Hartree-Fock approximation is also important in serving as a foundation for a variety of more accurate quantum chemical calculations that account for electron correlation. 

Special interest has been focused on the structure and reactivity of thiirane derivatives bearing exocyclic double bonds, methylenethiirane (or allene episulfide), thiiranone, and thiiranimine. The molecular structure of allene episulfide has been optimized by Closed-Shell SCF (CS-SCF) calculation using restricted Hartree-Fock (RHF) built into the GAUSSIAN 80 program, with the... 

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