whose determinant is shown below Isa() lsa(2) lsa(3) lsa( l$①ls阝(2)lsβ(3)1sB(4) psa()2a(2)2sx(3)2sa(4 s02sβ(2)2s阝(3)2s阝(4) Clearly, if one were to interchange any columns of this determinant, one changes the sigI of the function moreover if a determinant contains two or more rows that are identical (i.e, if one attempts to form such a function having two or more spin-orbitals equal),it vanishes. This is how such antisymmetric wave functions embody the Pauli exclusion principle a convenient way to write such a determinant is as follows Σp(-1)P中p1(1)中2(2)…中(N) where the sum is over all N! permutations of the N spin-orbitals and the notation (-1)p means that a-1 is affixed to any permutation that involves an odd number of pairwise interchanges of spin-orbitals and a +I sign is given to any that involves an even number To properly normalize such a determinental wave function, one must multiply it by (N! )2. So, the final result is that wave functions of the form
16 whose determinant is shown below Clearly, if one were to interchange any columns of this determinant, one changes the sign of the function. Moreover, if a determinant contains two or more rows that are identical (i.e., if one attempts to form such a function having two or more spin-orbitals equal), it vanishes. This is how such antisymmetric wave functions embody the Pauli exclusion principle. A convenient way to write such a determinant is as follows: SP (-1)p fP1 (1) fP2(2) … fPN(N), where the sum is over all N! permutations of the N spin-orbitals and the notation (-1)p means that a –1 is affixed to any permutation that involves an odd number of pairwise interchanges of spin-orbitals and a +1 sign is given to any that involves an even number. To properly normalize such a determinental wave function, one must multiply it by (N!)-1/2. So, the final result is that wave functions of the form 1sa(1) 1sa(2) 1sa(3) 1sa(4) 1sb (1) 1sb(2) 1sb(3) 1sb(4) 2sa(1) 2sa(2) 2sa(3) 2sa(4) 2sb (1) 2sb(2) 2sb(3) 2sb(4)
=(N)>p(-l)Pφp(1)φ2(2)….d(N) have the proper permutational antisymmetry. Note that such functions consist of as sum of N! factors, all of which have exactly the same number of electrons occupy ing the same number of spin orbitals; the only difference among the N! terms involves which electron occupies which spin-orbital. For example, in the 1sa2sa function appropriate to the excited state of He. one has 平=(2)"{1sa(1)2(2)-2sa(1)1sa(2)} This function is clearly odd under the interchange of the labels of the two electrons, yet each of its two components has one electron is a lsa spin- orbital and another electron in a 2sa spin-orbital Although having to make p antisymmetric appears to complicate matters significantly, it turns out that the Schrodinger equation appropriate to the spin-orbitals in such an antisymmetrized product wave function is nearly the same as the Hartree Schrodnger equation treated earlier. In fact the resultant equation h中={-h2/2mV2-Ze2r+∑k<(r)(e/rD)|(r)>}r) Σk<(r”)(e2/rrD)l(r)>ds(r)}=E3中r)
17 Y = (N!)-1/2 SP (-1)p fP1 (1) fP2(2) … fPN(N) have the proper permutational antisymmetry. Note that such functions consist of as sum of N! factors, all of which have exactly the same number of electrons occupying the same number of spin orbitals; the only difference among the N! terms involves which electron occupies which spin-orbital. For example, in the 1sa2sa function appropriate to the excited state of He, one has Y = (2)-1/2 {1sa(1) 2sa(2) – 2sa(1) 1sa(2)} This function is clearly odd under the interchange of the labels of the two electrons, yet each of its two components has one electron is a 1sa spin-orbital and another electron in a 2sa spin-orbital. Although having to make Y antisymmetric appears to complicate matters significantly, it turns out that the Schrödinger equation appropriate to the spin-orbitals in such an antisymmetrized product wave function is nearly the same as the Hartree Schrödnger equation treated earlier. In fact, the resultant equation is he fJ = {– h2 /2m Ñ 2 -Ze2 /r + SK <fK(r’) |(e2 /|r-r’|) | fK(r’)>} fJ (r) - SK <fK(r’) |(e2 /|r-r’|) | fJ (r’)> fK(r)} = eJ fJ (r)
In this expression, which is known as the Hartree-Fock equation, the same kinetic and nuclear attraction potentials occur as in the Hartree equation. Moreover, the same Σ∫φ(r)e/r-rlr)dr3=x<、(r)er-rlj(r)=Jk(r) appears. However, one also finds a so-called exchange contribution to the Hartree-Fock potential that is equal to∑<(r”)e2/r)|中(r”)(r) and is often written in short hand notation as 2 Kl, (r). Notice that the Coulomb and exchange terms cancel for the L=J case, this causes the artificial self-interaction term JLP(r) that can appear in the Hartree equations(unless one explicitly eliminates it) to automatically cancel with the exchange term Kl P(r)in the Hartree-Fock equations When the LCAO expansion of each Hartree-Fock(HF)spin-orbital is substituted into the above HF Schrodinger equation, a matrix equation is again obtained Σ<xlx>Cu=E1E<xx>C where the overlap integral <xulx> is as defined earlier, and the he matrix element <xhl x>=<xul-h12mVIx>+xu -/ Ix> EKn, CKn Csx(r)x()l(e/r-rDIxuux(r) x(r)(r)(e/r-r'DIx(r)(rP]
18 In this expression, which is known as the Hartree-Fock equation, the same kinetic and nuclear attraction potentials occur as in the Hartree equation. Moreover, the same Coulomb potential SK ò fK(r’) e2 /|r-r’| fK(r’) dr’ = SK <fK(r’)|e2 /|r-r’| |fK(r’)> = SK JK (r) appears. However, one also finds a so-called exchange contribution to the Hartree-Fock potential that is equal to SL <fL (r’) |(e2 /|r-r’|) | fJ (r’)> fL (r) and is often written in shorthand notation as SL KL fJ (r). Notice that the Coulomb and exchange terms cancel for the L=J case; this causes the artificial self-interaction term JL fL (r) that can appear in the Hartree equations (unless one explicitly eliminates it) to automatically cancel with the exchange term KL fL (r) in the Hartree-Fock equations. When the LCAO expansion of each Hartree-Fock (HF) spin-orbital is substituted into the above HF Schrödinger equation, a matrix equation is again obtained: Sm <cn |he | cm> CJ,m = eJ Sm <cn |cm> CJ,m where the overlap integral <cn |cm> is as defined earlier, and the he matrix element is <cn | he | cm> = <cn | – h2 /2m Ñ 2 |cm> + <cn | -Ze2 /|r |cm > + SK,h,g CK,h CK,g [<cn (r) ch (r’) |(e2 /|r-r’|) | cm (r) cg (r’)> - <cn (r) ch (r’) |(e2 /|r-r’|) | cg (r) cm (r’)>]
Clearly, the only difference between this expression and the corresponding result of Hartree theory is the presence of the last term, the exchange integral. The SCF interative procedure used to solve the Hartree equations is again used to solve the hf equations Next, I think it is useful to reflect on the physical meaning of the Coulomb and exchange interactions between pairs of orbitals. For example, the Coulomb integral J1. 2 J1, (rPe/r-r o(r)dr dr'appropriate to the two orbitals shown in Fig. 6.3 represents the Coulombic repulsion energy e/r-r of two charge densities, lo l and o2, integrated over all locations r and rof the two electrons p2(r) Overlap region Figure 6.3 An s and a p Orbital and Their Overlap Region
19 Clearly, the only difference between this expression and the corresponding result of Hartree theory is the presence of the last term, the exchange integral. The SCF interative procedure used to solve the Hartree equations is again used to solve the HF equations. Next, I think it is useful to reflect on the physical meaning of the Coulomb and exchange interactions between pairs of orbitals. For example, the Coulomb integral J1,2 = ò |f1 (r)|2 e 2 /|r-r’| f2 (r’)|2 dr dr’ appropriate to the two orbitals shown in Fig. 6.3 represents the Coulombic repulsion energy e2 /|r-r’| of two charge densities, |f1 | 2 and |f2 | 2 , integrated over all locations r and r’ of the two electrons. Figure 6.3 An s and a p Orbital and Their Overlap Region f1(r) f2(r') Overlap region
In contrast, the exchange integral K12=, (r)2(r)e /r-r'lp2 (r)P,r,)dr dr' can be thought of as the Coulombic repulsion between two electrons whose coordinates r and rare both distributed throughout the overlap region"o,2. This overlap region is where both o, and 2 have appreciable magnitude, so exchange integrals tend to be ignificant in magnitude only when the two orbitals involved have substantial regions of overlap Finally, a few words are in order about one of the most computer time-consuming parts of any Hartree-Fock calculation(or those discussed later)-the task of evaluating and transforming the two-electron integrals x(r)%(r)le7r-rDlx(r)x(r). Even when M GTOs are used as basis functions, the evaluation of M"/8 of these integrals poses a major hurdle. For example, with 500 basis orbitals, there will be of the order of 7. 8x10 such integrals. With each integral requiring 2 words of disk storage, this would require at least 1.5 x10 Mwords of disk storage. Even in the era of modern computers that possess 100 Gby disks, this is a significant requirement. One of the more important technical advances that is under much current development is the efficient calculation of such integrals when the product functions iu(r)i(r)andi,(r)%(r)that display the dependence on the two electrons coordinates r and r are spatially distant. In particular, multipolar expansions of these product functions are used to obtain more efficient approximations to their integrals when these functions are far apart. Moreover, such expansions offer a reliable way to"ignore"(i.., approximate as zero ) many integrals whose product functions are sufficiently distant. Such approaches show considerable promise for reducing the M/8 two-electron integral list to one whose size scales much
20 In contrast, the exchange integral K1,2 = ò f1 (r) f2 (r’) e2 /|r-r’| f2 (r) f1 (r’) dr dr’ can be thought of as the Coulombic repulsion between two electrons whose coordinates r and r’ are both distributed throughout the “overlap region” f1 f2 . This overlap region is where both f1 and f2 have appreciable magnitude, so exchange integrals tend to be significant in magnitude only when the two orbitals involved have substantial regions of overlap. Finally, a few words are in order about one of the most computer time-consuming parts of any Hartree-Fock calculation (or those discussed later)- the task of evaluating and transforming the two-electron integrals <cn (r) ch (r’) |(e2 /|r-r’|) | cm (r) cg (r’)>. Even when M GTOs are used as basis functions, the evaluation of M4 /8 of these integrals poses a major hurdle. For example, with 500 basis orbitals, there will be of the order of 7.8 x109 such integrals. With each integral requiring 2 words of disk storage, this would require at least 1.5 x104 Mwords of disk storage. Even in the era of modern computers that possess 100 Gby disks, this is a significant requirement. One of the more important technical advances that is under much current development is the efficient calculation of such integrals when the product functions cn (r) cm (r) and cg (r’) ch (r’) that display the dependence on the two electrons’ coordinates r and r’ are spatially distant. In particular, multipolar expansions of these product functions are used to obtain more efficient approximations to their integrals when these functions are far apart. Moreover, such expansions offer a reliable way to “ignore” (i.e., approximate as zero) many integrals whose product functions are sufficiently distant. Such approaches show considerable promise for reducing the M4 /8 two-electron integral list to one whose size scales much