Electronic integrals

However, F is a quartic fiinction of the -s because involves two-electron integrals (( i. ( ). tliat... 

The //yj matrices are, in practice, evaluated in temis of one- and two-electron integrals over the MOs using the Slater-Condon mles [M] or their equivalent. Prior to fomiing the Ffjj matrix elements, the one-and two-electron integrals. 

Both T and E are expressed in tenns of two-electron integrals (i,j m.,n ) coupling the virtual spin orbitals... 

Essentially all of the teclmiques discussed above require the evaluation of one- and two-electron integrals over the AO basis fiinctions (x l./lx ) and mentioned earlier, there are of the order of A /8... 

Once the requisite one- and two-electron integrals are available in the MO basis, the multiconfigurational wavefunction and energy calculation can begin. Each of these methods has its own approach to describing tlie configurations d),. j included m the calculation and how the C,.] amplitudes and the total energy E are to be... 

For all calculations, the choice of AO basis set must be made carefully, keeping in mind the scaling of the two-electron integral evaluation step and the scaling of the two-electron integral transfonuation step. Of course, basis fiinctions that describe the essence of the states to be studied are essential (e.g. Rydberg or anion states require diffuse functions and strained rings require polarization fiinctions). 

McMurchie L E and Davidson E R 1978 One-and two-electron integrals over Cartesian Gaussian functions J. Comp. Phys. 26 218-31 Gill P M W 1994 Molecular integrals over Gaussian basis functions Adv. Quantum Chem. 25 141-205... 

This is because no four-indexed two-electron integral like expressions enter into the integrals needed to compute the energy. All such integrals involve p(r) or the product p(/)p(r) because p is itself expanded in a basis (say of M functions), even the term p(r)p(r) scales no worse than tvF. The solution of the KS equations for the KS orbitals ([). involves solving a matrix eigenvalue problem this... 

Split valence basis sets generally give much better results than minimal ones, but at a cost. Remember that the number of two-electron integrals is proportional to kf , where W is the number of basis functions. Whereas STO-3G has only live ba.sis functions for carbon, 6-31G has nine, resulting in more than a tenfold increase in the size of the calculation,... 

MP2 correlation energy calculations may increase the computational lime because a tw o-electron integral Iran sfonnalion from atomic orbitals (.40 s) to molecular orbitals (MO s) is ret]uired. HyperClicrn rnayalso need additional main memory arul/orcxtra disk space to store the two-eleetron integrals of the MO s. 

Th e calcn lation of the two-electron repulsion mtegraism ah iniiio method is inevitable and time-consuming. The computational iim e is main ly dom in alcd by th e performance of Ih e two-electron integral calcii lalion. The following item s can con trol the performance of the two-electron integrals. 

The amount of computation for MP2 is determined by the partial tran si ormatioii of the two-electron integrals, what can be done in a time proportionally to m (m is the u umber of basis functions), which IS comparable to computations involved m one step of(iID (doubly-excitcil eon figuration interaction) calculation. fo save some computer time and space, the core orbitals are frequently omitted from MP calculations. For more details on perturbation theory please see A. S/abo and N. Ostlund, Modem Quantum (. hern-isir > Macmillan, Xew York, 198.5. 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

After considering the symmetry of the two-electron integrals, we have... 

III an SCF calculation. many iterations may beneetled to achieve SCr con vergeiice. In each iteration all the two-electron integrals are retrieved to form a Fock matrix. Fast algorith m s to retrieve the two-cicetron s integrals arc important. 

For fast access to the two-electron integrals, a fonr-dimensional array migh t be straigh t for ward. Th e four in times of the four diinen -sional array correspond to the four basis function indices, p, v, X, an d a. respectively. However, the four dimen sional array m ay lake a huge mam memtiry or computer disk space even for a mediiim-si/e molecule. Therefore, this may not be practical. 

RalTenetti [R. C. RalTenetli. Chem. I hys. Lett. 20, iiiS.bfl 97iS ) proposed another way to store the two-electron integrals in ah iniiio calculations. RatTenetti rewrote (93) on page 2.31 to read... 

So only the two-electron integrals wilh p. > v. and I>aand [p.v > 7.a need to he computed and stored. Dp.v.la on ly appears m Gpv, and Gvp, w hereas ih e original two-electron integrals con tribute to other matrix elemen is as well. So it is m iich easier to form ih e Fock matrix by using the siipermairix D and modified density matrix P th an the regular format of the tw O-electron in tegrals and stan dard den sity m atrix. 

Since the first formulation of the MO-LCAO finite basis approach to molecular Ilartree-Pock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — arc calculated and stored on external storage. Th e second step then con sists of the iterative solution of the Roothaan equations, where the integrals from the first step arc read once for every iteration. 

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