Closed-shell states

For closed-shell states, we found an energy expression e = Tr(Ph,)+ iTr(PG)... 

We satv earlier that the variational energy for a closed-shell state formed from electron configurations such as... 

In molecules with more than one unpaired electron, electron-electron interactions can have a significant role in the stabilization of the system. Bond formation that results from direct overlap is highly favorable and, thus is an overriding consideration in all low-spin polyradicals, even to the extent that the system sometimes adopts a strained, closed-shell state as opposed to a polyradical. In cases in which bonding cannot occur, indirect interactions that are usually insignificant, such as electron exchange and spin-polarization, can have significant impact. The presence of these interactions is often reflected in the thermochemical properties. 

The minimum of the triplet state corresponds to a larger Fe-O-O angle (131°) than the S = 1 state (121°), as we also found for a small FeP(Im)(02) system [6b]. On the other hand, an S = 0 closed-shell state is well separated in energy in the linear conformation (22 kcal mol-1 relative to the ground triplet state), but becomes very close to the ground state in the bent conformation (1.4 kcal mob1). Its electronic configuration can be shown schematically as ... 

The imposition of a closed shell state condition is not mandatory provided that the wavefunction of the state being sought can be written as in eq. (1). In this case, ( )o should be replaced by the dominant determinant ( ) and hence the eq. (1) should read as... 

In terms of these conditions, a fc-particle hierarchy of approximations can be defined, with Hartree-Fock as the one-particle approximation for closed-shell states. Unfortunately, the stationarity conditions do not determine the fully, and for their constmction additional information is required, which essentially guarantees -representability. Nevertheless, the fe-particle hierarchy based on the irreducible stationarity conditions opens a promising way for the solution of the -electron problem. 

The particle-hole formalism has been introduced as a simplihcation of many-body perturbation theory for closed-shell states, for which a single Slater determinant dominates and is hence privileged. One uses the labels i,j, k,... for spin orbitals occupied in <1> and a,b,c,... for spin orbitals unoccupied virtual) in . 

We expand the Hamiltonian and the IBCj in terms of a perturbation parameter p. in the spirit of M0ller-Plesset perturbation theory [34]. Details are found in Ref. [25]. We need not worry about the particle rank to which we have to go, since this is fully controlled by the perturbation expansion. We limit ourselves to a closed-shell state, such that the zeroth order is simply closed-shell Hartree-Fock. 

The calculated barrier for the cyclization of 33t to 29 is 6 kcal/mol after taking into consideration that the CASPT2 method overestimates the energy difference between open- and closed-shell states by 3 kcaFmol. The correspondingly adjusted predicted 6 kcal/mol barrier is in nearly exact agreement with the experimental value (5.6 0.3 kcal/mol). " ... 

We studied the hs S=2, the triplet S=l, the BS S=0 as well as the closed-shell state for this reactive intermediate and present the BP86/RI7TZVP structure optimizations in Fig. 4. Local-spin distributions are given in Table IV for local expectation values. 

We now consider the theoretical calculation of excited-state wave functions. This is more difficult than ground-state calculations because we are dealing with open-shell configurations. The Hartree-Fock equations for a state of an open-shell configuration have a more complicated form than for closed shells, and there exist close to a dozen different approaches to excited-state Hartree-Fock calculations. As noted earlier, the Hartree-Fock wave function for a closed-shell state is a single determinant, but for open-shell states, we may have to take a linear combination of a few Slater determinants to get a Hartree-Fock function that is an eigenfunction of S and Sz and has the correct spatial symmetry. 

For open-shell systems, an extension of the Hartree-Fock method, called the unrestricted Hartree-Fock method, is sometimes used. For a closed-shell state, the lowest Hartree-Fock energy is generally obtained... 

In conclusion, we note that experimental data, usually from infrared or ultraviolet spectroscopy, are often expressed by giving values of the parameters presented in equation (2.182). The formula is able to model low-lying vibrational levels of a molecule in a closed shell state quite accurately. 

SAC singlet closed-shell state (ground state)... 

We may recall that the desirability of ensuring size-extensivity for a closed-shell state was one of the principal motivations behind the formulation of the MBPT for the closed-shells. The linked cluster theorem of Bruckner/25/, Goldstone/26/ and Hubbard/27/, proving that each term in the perturbation series for energy can be represented by a linked (connected) diagram directly reflects the size-extensivity of the theory. Hubbard/27/ and Coester/30/ even pointed out immediately after the inception of MBPT/25,26/, that the size-extensivity is intimately related to a cluster expansion structure of the associated wave-operator that is not just confined only to perturbative theory. The corresponding non-perturbative scheme for the closed-shells was first described by Coester and Kummel/30,31/ in nuclear physics and this was transcribed to quantum chemistry... 

For an N-electron closed-shell state with a dominant reference function, the exact wave-function 4 can be written as a superposition of various n-fold excited determinants on 4.4 may thought to be generated from by anwave operator... 

For closed-shell states, there is, according to Eq. 59, no spin contribution to the paramagnetic part of the magnetizability, the only contribution coming from the orbital motion of the electrons ... 

Since, for closed-shell states, spin interactions do not contribute to any of our properties, we shall here ignore the contributions from the spin degrees of freedom and use the simpler spin-0 Hamiltonian of Eq. 69. 

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