Hartree-Fock wave functions, restricted

If an excited state is concerned, this is done under the restriction that the function should be orthogonal to all of the lower-energy states. We may specify these as the uni-configurational Hartree-Fock wave functions . The "best orbitals constructing the determinants in these wave functions are in general not orthogonal to each other. 

Molecular orbitals from linear combination of atom orbitals Spin-restricted Hartree-Fock. Wave function constructed from antisymmetrized product of doubly occupied spin orbitals (UHF, spin-unrestricted Hartree-Fock calculations are used for excited states and radicals)... 

Scattered electron intensities have been calculated for all 27 normal modes of the propane molecule. The electronic ground state of the molecule was described by the restricted Hartree-Fock wave function within Sadlej s basis set23. Cross sections obtained from DMR calculations were converted to a theoretical band spectrum by assuming Lorentzian shape for all bands with a half-width taken from the width of the observed elastic peak. The bands were centered at the positions of observed vibrational frequencies. 

For Eq. (11) S is the Bragg vector S = 2ttH, IT is the row vector (htk,l) and the scalar S - S = 4ir sin 0/A. The index / covers the N atoms in the unit cell. The atomic scattering factor f (S) is the Fourier-Bessel transform of the electronic, radial density function of the isolated atom. This density function is usually derived from a spin-restricted Hartree-Fock wave function for the atom in its ground state. The structure fac-... 

The question for a more systematic inclusion of electronic correlation brings us back to the realm of molecular quantum chemistry [51,182]. Recall that (see Section 2.11.3) the exact solution (configuration interaction. Cl) is found on the basis of the self-consistent Hartree-Fock wave function, namely by the excitation of the electrons into the virtual, unoccupied molecular orbitals. Unfortunately, the ultimate goal oi full Cl is obtainable for very small systems only, and restricted Cl is size-inconsistent the amount of electron correlation depends on the size of the system (Section 2.11.3). Thus, size-consistent but perturbative approaches (Moller-Plesset theory) are often used, and the simplest practical procedure (of second order, thus dubbed MP2 [129]) already scales with the fifth order of the system s size N, in contrast to Hartree-Fock theory ( N ). The accuracy of these methods may be systematically improved by going up to higher orders but this makes the calculations even more expensive and slow (MP3 N, MP4 N ). Fortunately, restricted Cl can be mathematically rephrased in the form of the so-called coupled clus-... 

SCF MO Wave Functions for Open-Shell States. For SCF MO calculations on closed-shell states of molecules and atoms, electrons paired with each other are almost always given precisely the same spatial orbital function. A Hartree-Fock wave function in which electrons whose spins are paired occupy the same spatial orbital is called a restricted Hartree-Fock (RHF) wave function. (The unmodified term Hartree-Fock wave function is understood to mean the RHF wave function.)... 

So fiir in this chapter we have discussed the Hartree-Fock equations from a formal point of view in terms of a general set of spin orbitals xj. We are now in a position to consider the actual calculation of Hartree-Fock wave functions, and we must be more specific about the form of the spin orbitals. In the last chapter we briefly discussed two types of spin orbitals restricted spin orbitals, which are constrained to have the same spatial function for a (spin up) and jS (spin down) spin functions and unrestricted spin orbitals, which have different spatial functions for a and P spins. Later in this chapter we will discuss the unrestricted Hartree-Fock formalism and unrestricted calculations. In this section we are concerned with procedures for calculating restricted Hartree-Fock wave functions and, specifically, we consider here... 

With the background of the previous sections we are now in a position to describe the actual computational procedure for obtaining restricted closed-shell Hartree-Fock wave functions for molecules, i.e., wave functions Fo>-Some authors restrict the term Hartree-Fock solution to one that is at the... 

Hartree-Fock dieory in special cases, such as for restricted open-shell wave functions, involves a multideterminantal wave function. Since we wiU be concerned only with unrestricted open-shell wave functions, all Hartree-Fock wave functions will be single determinants. 

Abstract An expression for the square of the spin operator expectation valne, S, is obtained for a general complex Hartree-Fock wave fnnction and decomposed into four contributions the main one whose expression is formally identical to the restricted (open-shell) Hartree-Fock expression. A spin contamination one formally analogous to that found for spin nnrestricted Hartree-Fock wave functions. A noncollinearity contribntion related to the fact that the wave fnnction is not an eigenfunction of the spin- S operator. A perpendicularity contribution related to the fact that the spin density is not constrained to be zero in the xy-plane. All these contributions are evaluated and compared for the H2O+ system. The optimization of the collinearity axis is also considered. 

In this section, we consider spin-unrestricted MPPT, taking as our unperturbed state the spin-unrestricted Hartree-Fock wave function and as our zero-order Hamiltonian the Fock operator. A spin-restricted treatment suitable for closed-shell states is given in Section 14.4, following the discussion of CCPT in Section 14.3. 

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. 

The purpose of the present chapter is to discuss the structure and construction of restricted Hartree-Fock wave functions. We cover not only the traditional methods of optimization, based on the diagonalization of the Fock matrix, but also second-order methods of optimization, based on an expansion of the Hartree-Fock eneigy in nonredundant orbital rotations, as well as density-based methods, required for the efficient application of Hartree-Fock theory to large molecular systems. In addition, some important aspects of the Hartree-Fock model are analysed, such as the size-extensivity of the energy, symmetry constraints and symmetry-broken solutions, and the interpretation of orbital energies in the canonical representation. 

Quantum mechanics calculations use either of two forms of the wave function Restricted Hartree-Fock (RHF) or Unrestricted Hartree-Fock (UHF). Use the RHF wave function for singlet electronic states, such as the ground states of stable organic molecules. 

The classification of CSFs into single, double,... excitations is straightforward and unambiguous for a closed-shell Hartree-Fock reference function. In open-shell or multireference cases, there are more possibilities for defining these excited CSFs, some of which interact with the reference space in the lowest order of perturbation theory, and some of which do not. It is very common to exclude the latter excitations. This is termed restricting the wave function to the first-order interacting space. ... 

As discussed in the present section and also in Sections 10.5 and 10.6, canonical Hartree-Fock theory offers many advantages. However, for open-shell RHF states, this approach is not always possible because of the constraints imposed by the spin- and space-symmetry adaptation of the wave fiinction. In such cases, the wave function must be calculated using the more general methods of Section 10.8 in the present section, we restrict our attention to closed-shell systems. In addition, the application of canonical Hartree-Fock theory is restricted to small and moderately large systems since it requires an amount of work that scales at least cubically with the size of the system for large systems, the methods of Section 10.7 must be used instead. 

Direct dynamics attempts to break this bottleneck in the study of MD, retaining the accuracy of the full electronic PES without the need for an analytic fit of data. The first studies in this field used semiclassical methods with semiempirical [66,67] or simple Hartree-Fock [68] wave functions to heat the electrons. These first studies used what is called BO dynamics, evaluating the PES at each step from the elech onic wave function obtained by solution of the electronic structure problem. An alternative, the Ehrenfest dynamics method, is to propagate the electronic wave function at the same time as the nuclei. Although early direct dynamics studies using this method [69-71] restricted themselves to adiabatic problems, the method can incorporate non-adiabatic effects directly in the electionic wave function. 

Another way of constructing wave functions for open-shell molecules is the restricted open shell Hartree-Fock method (ROHF). In this method, the paired electrons share the same spatial orbital thus, there is no spin contamination. The ROHF technique is more difficult to implement than UHF and may require slightly more CPU time to execute. ROHF is primarily used for cases where spin contamination is large using UHF. 

A UHF wave function may also be a necessary description when the effects of spin polarization are required. As discussed in Differences Between INDO and UNDO, a Restricted Hartree-Fock description will not properly describe a situation such as the methyl radical. The unpaired electron in this molecule occupies a p-orbital with a node in the plane of the molecule. When an RHF description is used (all the s orbitals have paired electrons), then no spin density exists anywhere in the s system. With a UHF description, however, the spin-up electron in the p-orbital interacts differently with spin-up and spin-down electrons in the s system and the s-orbitals become spatially separate for spin-up and spin-down electrons with resultant spin density in the s system. 

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. 

In the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

The Hartree-Fock description of the hydrogen molecule requires two spinorbitals, which are used to build the single-determinant two-electron wave function. In the Restricted Hartree-Fock method (RHF) these two spinorbitals are created from the same spatial... 

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