Closed shell SCF calculation

At this point we have developed a method for handling symmetric one-electron operators a useful step, but far from adequate to cover all cases we are interested in, since we have said nothing yet about the much more demanding case of two-electron integrals. Let us consider first building the Fock matrix for a closed-shell SCF calculation. This is perhaps most easily expressed in terms of a P supermatrix with elements... 

Ethylene, HF and HsO+ have been used as a model system in the ab initio closed-shell SCF calculation of the acid-catalysed hydrogenation of olefins. While catalysis by HF exhibits bifunctional character, catalysis by HsO+ proceeds via initial formation of a carbocation72. 

M. Dupuis and H. F. King, Int. ]. Qttantum Chem., 11, 613 (1977). Molecular Symmetry and Closed-Shell SCF Calculations. 

We have thus established the basis set to use and the active space for the CASSCF calculations. To perform the CASSCF calculations we need in addition a set of starting orbitals. These are most easily obtained from an SCF calculation. If we have access only to a closed shell SCF program we can... 

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... 

Two types of Cl calculation are presented. The relaxed Cl calculations employ the neutral SCF orbitals in the calculation of the ground-state energy of the neutral, and the ion SCF orbitals are used for the ion ground-state energy. The unrelaxed Cl calculations use a common set of orbitals for both the neutral and ion calculations. The orbitals are taken from a ground-state SCF calculation for whichever of the two is the closed-shell system. The unrelaxed Cl calculations are similar in this respect to the EOM calculations, which utilize only one set of closed-shell SCF orbitals throughout. Comparison of the relaxed and unrelaxed Cl calculations also affords a check of the Cl convergence. A number of different relaxed and unrelaxed Cl calculations are made for each system. The simplest involves... 

The solution of the simultaneous equations (32) requires some care because of their size. The number of independent elements of the matrix (/ which describes the perturbed orbitals is, for closed-shell SCF wavefunctions, the product of the number of occupied and the number of virtual molecular orbitals. This can be a large number—several thousand for a calculation on a big molecule. This makes it difficult to solve (32) by the conventional techniques used for small sets of simultaneous equations. However, an iterative approach introduced by Pople et al. provides the answer. This method constructs the solution 1/ as a linear combination of trial vectors... 

Most of the general features discussed above for closed-shell SCF will also apply to calculations using other types of wavefunction. For other varieties of... 

In Section 3.7 we perform a number of ab initio calculations to illustrate the application and the results of the closed-shell SCF procedure. Our principal aim is to give the reader a feeling for a few of the problems to which ab initio SCF calculations can be applied, and the accuracy that can be expected of such Hartree-Fock calculations. To systematize these applications, we apply a standard hierarchy of basis sets to each problem. 

Static and frequency dependent dipole polarizabilities can be calculated for closed shell SCF, open shell UHF, and for closed shell DFT cases. ... 

Show how the method of Werner et al. (p. 273) may be applied in simple closed-shell SCF theory in order to obtain a rapidly convergent procedure. How would you implement the method of Section 8.3 (singular-value decomposition) in an actual calculation [Hint Start from an energy expression (e.g. (5.3.18)) identify the density-matrix elements (e.g. P = 2b , r, s occupied, = 0, otherwise) substitute in (8.4.12) and (8.4.13), and find the various elements of B.]... 

To appreciate another problem with highly diffuse basis sets, one must first understand that it is common practice to use MOs from the neutral molecule - call it M, as above - as an initial guess for the SCF calculation on M . This choice can greatly accelerate the anion calculation since the closed-shell SCF cycles are 50% less expensive and often provide an excellent guess if the anion is weakly-bound and only slightly perturbs the MOs of the neutral molecule. Moreover, insofar as both the M and M calculations are required to compute the VDE, nothing is lost by first computing the SCF wave function for M. However, for (H20) isomers with very small VDEs... 

The energy-gradient method was first put into practice for closed-shell HF calculations, but it has recently been extended to spin-unrestricted and restricted-HF calculations on open-shell systems and to certain types of multiconfigurational SCF calculations. A method has been developed, based on the perturbed HF theory, to calculate the derivatives of MO coefficients. With this in hand, one is now able to calculate the energy gradient for a... 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

SCF-CI calculations were performed at 20 different intemuclear separations, from 1.2 bohr to 4-00. The lowest separate atom states are, B( P,2p) and H( S) therefore, in order to have a homolytic dissociation and three degenerate 2p orbitals on B we have adopted the closed shell Fock hamiltonian with fractional occupation [23] one electron was placed in the 3(t orbital, correlating with H(ls) at infinite separation, and 1/3 each in the 4it and Itr orbitals correlating with B(2p). 

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