Energy of closed-shell system

For all higher-order correlation energies (of closed-shell systems), moreover, we can follow similar lines and evaluate the vacuum expectation values EI"I = (0c HX2 " 0c). This results in the standard Moeller-Plesset expressions for the energy corrections. 

MP2 energies of closed-shell systems, including an R12 correlation factor and using an auxilary basis set, are supported. 

J. Cizek and J. Paldus, A direct calculation of the excitation energies of closed-shell systems using the Green function technique, Int. J. Quantum Chem. Symp. 6 435 (1972) discussion of the status of large-molecule quantum chemistry, in Energy, Structure, and Reactivity , D. W. Smith and W. B. McRae, eds., Wiley, New York (1973), p. 389. 

Gutowski M, Piela L (1988) Interpretation of the Hartree-Fock interaction energy between closed-shell systems. Mol Phys 64 337-355... 

The latter expression clearly shows that Hartree-Fock wave functions are not properly correlated they allow two electrons of opposite spin to simultaneously occupy a same elementary volume of an atomic or molecular space. Consequently, two-electron properties which are completely determined by the second-order density matrix cannot be correctly evaluated at the Hartree-Fock level and, a fortiori, from approximate SCF wave functions. On the contrary, satisfactory values of one-electron properties may be generally provided by those functions, at least in the case of closed-shell systems. However, due to the large contribution of pair correlation, the energy changes associated with the so-called isodes-mic processes (Hehre et al., 1970) can be reasonably well predicted at the Hartree-Fock level and also using SCF wave functions. Indeed, in that case, correlation errors approximately balance each other. 

M. Gutowski and L. Piela, Mol. Phys., 64, 337 (1988). Interpretation of the Hartree-Fock Energy Between Closed-Shell Systems. 

The UHF energy is expected to be lower than the RHF one, but in the case of closed shell systems close to the equilibium geometry, the UHF energy usually coincides with the RHF one. This feature produces a sort of discontinuity in the potential energy curve the internuclear distance when increases. 

Ultraviolet photoelectron spectroscopy (UPS) involves ionization of valence electrons which probes metal-ligand bonding and change with ionization (from resonance effects as photon energy is scanned through metal M edge), thus directly studying redox processes in metal complexes allows for study of closed shell systems which are inaccessible via electronic absorption spectroscopy ... 

Every term in this expansion (as indicated by the integer at the end of the line) represents a Feyman-Goldstone diagram and could be drawn also graphically. These are the (three) well-known diagrams of the second-order correlation energy for closed-shell systems as found at different places in the literature. If we assume a Hartree-Fock (one-particle) spectrum, instead,... 

At the Hartree-Fock level (Hartree 1928 Fock 1930), the energies for closed-shell systems are evaluated using the restricted Hartree-Fock (RHF) method (Hall and Lennard-Jones 1951 Roothaan 1951). For the open-shell molecules, there are several methods that are available in most programs the unrestricted Hartree-Fock (UHF) method (Pople and Nesbet 1954), several variants of the restricted open-shell Hartree-Fock (ROHF) method (Hsu et al. 1976 McWeeny and Diercksen 1968), and the generalized valence bond (GVB) method (Bobrowicz and Schaefer 1977). 

The EOM-CCSD method has been shown to be an accurate approach for calculating excitation energies (EE-) (29,30), electron affinities (EA-) (31), and ionization potentials (IP-) (32) of closed-shell systems. For the computations presented here, a coupled-cluster singles and doubles (CCSD) calculation is first performed on the closed-shell NO3" anion system. The Hamiltonian is similarity... 

The sum of the zeroth-order and first-order energies thus corresponds to the Hartree-Fock energy (compare with Equation (2.110), which gives the equivalent result for a closed-shell 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. 

The F matrix elements in eqs. (15) and (16) are formally the same as for closed-shell systems, the only difference being the definition of the density matrix in eq. (17), where the singly occupied orbital (m) has also to be taken into account. The total electronic energy (not including core-core repulsions) is given by... 

These methods can give us useful information on radicals in a manner similar to that for closed-shell systems, provided the exploitation is correct. Of course, in expressions for total energy, bond orders, etc., a singly occupied orbital must be taken into account. One should be aware of areas where the simple methods give qualitatively incorrect pictures. The HMO method, for example, cannot estimate negative spin densities or disproportionation equilibria. On the other hand, esr spectra of thousands of radicals and radical ions have been interpreted successfully with HMO. On the basis of HMO orbital energies and MO symmetry... 

A detailed diseussion of these and other variants was given in (Ref.35). Attention must be called to the fact that these methods are not variational which causes the energies obtained with them to be lower than those obtained with the FCI method. The counterpart to this deffect is that excited states, open-shell systems, and radicals, can be calculated with as much ease as the ground state and closed-shell systems. Also, the size of the calculation is determined solely by the size of the Hilbert subspace chosen and does not depend in principle on the number of electrons since all happens as if only two electrons were considered. 

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