Detailed derivation of Gaussian orbital based matrix elements in electron structure calculations.

Size: px

Start display at page:

Download "Detailed derivation of Gaussian orbital based matrix elements in electron structure calculations."

Jodie Potter
6 years ago
Views:

1 Detailed derivation of Gaussian orbital based matri elements in electron structure calculations. T Petersson and B Hellsing Department of Physics, Väjö University, SE Väjö, Sweden Department of Physics, Göteborg University, SE-4 96 Göteborg, Sweden tomas.petersson@vu.se Abstract. A detailed derivation of the analytic solutions is presented for overlap, kinetic, nuclear attraction and electron repulsion integrals involving Cartesian Gaussian type orbitals. It is demonstrated how s-type orbitals can be used to evaluate integrals with higher angular momentum via the properties of Hermite polynomials and differentiation with respect to non-integration variables. PACS numbers:.3.rz, 3.5-p

2 Detailed derivation of Gaussian orbital based matri elements. Introduction In quantum chemistry, the Hartree-Fock (HF) approach is quite common in calculations of the electronic structure of molecules. Usually, one constructs the molecular orbitals by a linear combination of atomic orbitals, i.e., the molecular orbital is a sum of atomic orbitals, with different centers. The most common basis functions are the Slater type and the Gaussian-type functions. The use of Gaussian type functions was proposed by Boys [] since these are simpler to integrate, but the drawback is that one needs a larger basis set to achieve the same accuracy as with Slater type functions. In a HF calculation, the solution is found iteratively in a self-consistent procedure and various integrals have to be calculated in order to solve the equations. Here, we show how analytic formulas for these integrals can be obtained. Books in quantum chemistry [, 3] or computational physics [4] describing HF calculations seldom give any eamples beyond s-type functions, which are the easiest to calculate, and therefore contain no formulas for higher angular orbitals. Clementi and Davis [5] published the analytical result of the integrals involved, but for a novice reader it is hard to see through printing errors and the article does not present any derivations. Our derivations differ from those of Clementi and Davis. We believe the path we follow to be a slightly more pedagogical one, and that students in a more advanced quantum mechanics or computational physics course, in which the many electron problem is attacked computationally, may find this article useful. A Cartesian Gaussian type orbital (GTO), centered on atom A with orbital eponent α and non-negative integers l, m and n, is defined as φ A (r) = N( A ) l (y A y ) m (z A z ) n e α(r A) () with the normalization constant ( ) 3/4 α (8α) l+m+n / l! m! n! N =. () π (l)! (m)! (n)! When l = m = n =, we have an s-type GTO, when l + m + n = we have a p-type GTO, and so on. The word Gaussian function is used for a s-type GTO. A general idea to solve the integrals is given in section. Section 3 describes how the analytic results are derived, and the complete formulas for the integrals are written out in section 4. A suggestion for how to derive other solutions is found in section 5.. General outline of the problem It is simple to integrate a Gaussian function, but it is much more difficult to do it for a p-type or higher GTO. The idea is to generate matri elements involving orbitals with higher order GTO s from Gaussian functions without involving integration variables. This means that we can first integrate a product of Gaussian functions, and then apply an operator to get the desired result. The appropriate choice is to use Hermite polynomials since they are intimately connected to Gaussian functions. The Hermite

3 Detailed derivation of Gaussian orbital based matri elements 3 polynomials can be epanded as H n () = n! [ n] m= ( ) m () n m m! (n m)! the bracket-notation [k] is used to denote the largest integer k. The epansion of a monomial in Hermite polynomial reads n = n! [ n] H n i () n i! (n i)!. (4) i= With the use of Rodrigues formula H n () = ( )n e d n d n e the -component of an arbitrary GTO may be written as This relation gives us ( A ) l e α( A) = l! l d ( A ) l e α( A) = l! l [ l] i= [ l] i= i! (l i)! α l i l i l i i! (l i)! α l i A l i A l i (3) (5) e α( A). (6) d e α( A). (7) By taking the derivative over atomic position, A, instead of the usual position, r, the goal is achieved to epress an arbitrary GTO into a form the integration is only performed over a Gaussian function, i.e. s-type GTO. 3. The derivations We show now how one derives the analytic result for the overlap, kinetic, nuclear attraction and electron repulsion integrals for arbitrary GTO s from integrations over s-type GTO s. 3.. The overlap integral The overlap integral is defined as A B = d 3 r φ A (r)φ B (r) (8) φ A (r) and φ B (r) are defined as in (). As stated in (7), only Gaussian functions need to be integrated. reduces to solve d 3 r e α (r A) e α (r B) = e ηp(a B) d 3 r e γp(r P ) The problem = π 3/ γ 3/ p e ηp(a B) (9)

4 Detailed derivation of Gaussian orbital based matri elements 4 with η p = α α α + α γ p = α + α P = γ p (α A + α B) () and a special case of the Gaussian product theorem has been used, that is e α (r A) e α (r B) = e ηp(a B) e γp(r P ). () With the aid of (6), the -component of the unnormalized analytic result becomes A B = = l! l! l +l d ( A ) l e α ( A ) ( B ) l e α ( B ) [ l ] [ l ] i = i = l i l i A l i B l i [ i! i! (l i )! (l i )! α l i α l i ] e ηp(a B) () and similarly for the y- and z-components. It remains to establish the derivatives with respect to the atom positions, which is easily done with the help of Rodrigues formula (5), and the epansion of the Hermite polynomials (3). For the -component one obtains j A j j B j e ηp(a B) = ( ) j η j [ p H j ηp (A B ) ] e ηp(a B) [ j] = ( ) j j! r= ( ) r j r η j r p (A B ) j r e ηp(a B) (3) r! (j r)! j = j + j. (4) 3.. The kinetic integral The kinetic integral is defined as A B = d 3 r φ A (r) [ ] φ B (r). (5) In order to determine this integral, we are interested in the result of applying the Laplace operator to φ B. We restrict ourselves to the -component: [ ( B ) l e α ( B ) = α (l + )( B ) l α( B ) l + ] l (l )( B ) l e α ( B ). (6) The problem reduces to a sum of three overlap integrals with different angular momenta.

5 Detailed derivation of Gaussian orbital based matri elements The nuclear attraction integral The nuclear attraction integral is defined as A Z c r c B = Z c d 3 r φ A(r)φ B (r). (7) r r c The denominator can be taken care of by using, for eample, a Fourier transform, but a shorter way to the same result is the relation d e a = π (8) a which gives r r c = du e u (r r c). (9) π In order to solve the nuclear attraction integral with Gaussian functions, the Gaussian product theorem () is applied Z c d 3 r r r c e α (r A) e α (r B) = Z c d 3 r π = Z c e ηp(a B) d 3 r π du e α (r A) α (r B) u (r r c) = Z c e ηp(a B) du e γpp u rc π du e γp(r P ) u (r r c) d 3 r e (γp+u )r +(γ pp +u r c) r () the variables η p, γ p and P are as in (). In order to solve the last integral in (), the relation ( ) π dt e (at +bt+c) b = erf e (b ac)/a () 4a a is used to arrive at dt e (at bt) = π a eb /a. () We identify a = γ p + u and b = γ p P + u r c. The result for the nuclear attraction integral with Gaussian functions, after integration over r, will be With a change of variables πz c e ηp(a B) du (γ p + u ) 3/ e γpu (P r c) /(γ p+u ). (3) t = u (γ p + u ) du = (γ p + u ) 3/ γ p dt, (4) the end result for the nuclear attraction integral with Gaussian functions becomes πz c e ηp(a B) γ p dt e γp(p rc) t. (5)

6 Detailed derivation of Gaussian orbital based matri elements 6 The result is similar to (), but with two functions to differentiate. The solution is to apply Leibniz s theorem for differentiation of a product ( ) n n (uv) = n r u n r v. (6) n r r n r r= The net step is to establish the derivatives with respect to the atom positions. The result from differentiation of e ηp(a B) is found in (3). For the -component, the differentiation of the integral becomes, j A j j B j [ j] dt e γp(p rc) t = α j α j ( ) j+u j u (P r c ) j u j! u! (j u)! γp u u= dt t (j u) e γp(p rc) t (7) j = j + j. (8) The same integral in the right-hand side of (7) will also appear for the y- and z- components. The integral can be evaluated by or by the recursion F ν (u) = dt t ν e ut [ π = (ν)! ν! 4 ν u erf u e u ν+/ ν k= ] (ν k)! 4 k (ν k)! u k+ F ν (u) = uf ν+(u) + e u (3) ν + which is only stable numerically if it is performed downward due to round-off errors. Thus, (7) can be written j A j j B j [ j] dt e γp(p rc) t = α j α j ( ) j+u j u (P r c ) j u j! u! (j u)! γp u 3.4. The electron repulsion integral u= The electron repulsion integral is defined as A, C r B, D = (9) F j u ( γp (P r c ) ). (3) d 3 r d 3 φ A (r )φ B (r )φ C (r )φ D (r ) r. (3) r r We take care of the denominator in a similar way as in (9). With the use of the Gaussian product theorem () and some rewriting in order to take advantage of (),

7 Detailed derivation of Gaussian orbital based matri elements 7 the electron repulsion integral with Gaussian functions will be d 3 r d 3 r r r e α (r A) e α (r B) e α 3(r C) e α 4(r D) = e ηp(a B) e ηq(c D) d 3 r d 3 r du π e u r γpp γ q(r Q) e [(γp+u )r (γpp +u r ) r ] (33) η p = α α η q = α 3α 4 α + α α 3 + α 4 γ p = α + α γ q = α 3 + α 4 P = γ p (α A + α B) Q = γ q (α 3 C + α 4 D). (34) We identify a = γ p + u and b = γ p P + u r, use () and get for the electron repulsion integral with Gaussian functions, after integration over r πe ηp(a B) e ηq(c D) d 3 r du (γ p + u ) 3/ e (γpp +u r ) /(γ p+u ) e u r γpp γ q(r Q). (35) We rearrange the terms in order to integrate over r, identifies the terms a = [γ p γ q + (γ p + γ q )u ]/(γ p + u ), b = [γ p u P + (γ p + u )γ q Q]/(γ p + u ) to get the form of (). The result for the electron repulsion integral with Gaussian functions then becomes π 5/ e ηp(a B)e ηq(c D) du (η + u ) 3/ e ηu (P Q) /(η+u ) (36) (γ p + γ q ) 3/ η = With a change of variables t = γ pγ q γ p + γ q. (37) u du = (η + u ) 3/ η + u η dt (38) the end-result for the electron repulsion integral with Gaussian functions becomes π 5/ e ηp(a B)e ηq(c D) γ p γ q (γ p + γ q ) / dt e η(p Q) t. (39) The result from differentiation of e ηp(a B) is found in (3), and the differentiation of e ηq(c D) is similar. The differentiation of the integral becomes, for the -component, j A j j B j j 3 C j 3 = αj α j α j 3 3 α j 4 4 γ j +j p γ j 3+j 4 q j 4 D j 4 [ j] j! u= dt e η(p Q) t ( ) u+j +j (P Q ) j u η j u ( F ) j u η(p Q) (4) u! (j u)!

8 Detailed derivation of Gaussian orbital based matri elements 8 j = j + j + j 3 + j 4. (4) The function F ν (η(p Q) ) of (4) will also show up for the y- and z-components. 4. The complete analytic solutions for the integrals We now write the complete analytical result for each integral considered, by using the results from the last section. The Cartesian GTO used here are defined as φ A (r) = N ( A ) l (y A y ) m (z A z ) n e α (r A) (4) φ B (r) = N ( B ) l (y B y ) m (z B z ) n e α (r B) (43) φ C (r) = N 3 ( C ) l 3 (y C y ) m 3 (z C z ) n 3 e α 3(r C) (44) φ D (r) = N 4 ( D ) l 4 (y D y ) m 4 (z D z ) n 4 e α 4(r D) (45) N i = ( αi π (46) ) 3/4 [ (8αi ) l i+m i +n i ] / l i! m i! n i! (47) (l i )! (m i )! (n i )! r R = ( R )î + (y R y )ĵ + (z R z )ˆk. (48) The bracket-notation [k] for summations, is used to denote the largest integer k. 4.. The overlap integral The complete analytic solution to the overlap integral now reads ( ) 3/ π A B = N N e ηp(a B) S γ p S z (49) i,i,o j,j,p S y k,k,q γ p = α + α η p = α α γ p (5) i,i,o S = ( )l l! l! γ l +l p i γ(i +i )+o p (A B ) Ω o (l i )! (l i )! (Ω o)! i o ( ) o Ω! α l i i o α l i i o 4 i +i +o i! i! o! (5) Ω = l + l (i + i ). (5) S y and S z are similarly defined in terms of the y- and z-components. The summations are over the following ranges: i = l i = l o = Ω (53) and analogously for the y- and z-components.

9 Detailed derivation of Gaussian orbital based matri elements The kinetic integral The complete analytic solution to the kinetic integral is A B = N N [α (4(l + m + n ) + 6) A B 4α { A B, l + + A B, m + + A B, n + } l (l ) A B, l m (m ) A B, m ] n (n ) A B, n (54) B, l + = ( B ) l + (y B y ) m (z B z ) n e α (r B) (55) and similarly for the other terms. unnormalized functions The nuclear attraction integral The overlap integrals should be performed with The complete analytic solution to the nuclear attraction integral is A Z c B = Z cn N π e ( ηp(a B) A A y A z F ν γp (P r c ) ) (56) r c γ p i,i o,o r,u j,j p,p s,v k,k q,q t,w γ p = α + α η p = α + α γ p P = γ p (α A + α B) (57) A = ( ) l +l l! l! i i,i o,o r,u ( ) o+r (o + o )! 4 i +i +r i! i! o! o! r! i α o i r α o i r (A B ) o +o r (l i o )! (l i o )! (o + o r)! ( ) u µ! (P r c ) µ u 4 u u! (µ u u)! γ o +o r+u p µ = l + l (i + i ) (o + o ) o o r (58) ν = µ + µ y + µ z (u + v + w). (59) A y and A z are similarly defined in terms of the y- and z-components. The summations are over the following ranges: i = l i = l o = l i o = l i r = (o + o ) u = µ (6) and similarly for the y- and z-components. The function F ν (γ p (P r c ) ) is evaluated as in (9).

10 Detailed derivation of Gaussian orbital based matri elements 4.4. The electron repulsion integral The complete analytic solution to the electron repulsion integral is A, C B, D = N N N 3 N 4 π 5/ e ηp(a B) e ηq(c D) r γ p γ q γp + γ q ( J z F ) ν η(p Q) (6) J J y i,i,i 3,i 4 o,o,o 3,o 4 r,r,u j,j,j 3,j 4 p,p,p 3,p 4 s,s,v k,k,k 3,k 4 q,q,q 3,q 4 t,t,w γ p = α + α γ q = α 3 + α 4 P = γ p (α A + α B) i,i,i 3,i 4 o,o,o 3,o 4 r,r,u η p = α + α η q = α 3 + α 4 Q = (α 3 C + α 4 D) γ p γ q γ q η = γ pγ q (6) γ p + γ q J = ( )l +l l! l! γ l +l p i i o o ( )o+r (o + o )! 4 i +i +r i! i r! o! o! r! αo i r α o i r γ (i +i )+r p (A B ) o +o r (l i o )! (l i o )! (o + o r )! l 3! l 4! γ l 3+l 4 q i 3 i 4 o 3 o 4 ( )o3+r (o 3 + o 4 )! 4 i 3+i 4 +r i3! i r 4! o 3! o 4! r! αo 4 i 3 r 3 α o 3 i 4 r 4 γ (i 3+i 4 )+r q (C D ) o 3+o 4 r (l 3 i 3 o 3 )! (l 4 i 4 o 4 )! (o 3 + o 4 r )! ( ) u µ! η µ u (P Q ) µ u 4 u u! (µ u u)! µ = l + l + l 3 + l 4 (i + i + i 3 + i 4 ) (o + o + o 3 + o 4 ) (63) ν = µ + µ y + µ z (u + v + w). (64) J y and J z are similarly defined in terms of the y- and z-components. The summations are over the following ranges: i = l i = l i 3 = l 3 i 4 = l 4 o = l i o = l i o 3 = l 3 i 3 o 4 = l 4 i [ 4 ] r = (o + o ) r = (o 3 + o 4 ) u = µ (65)

11 Detailed derivation of Gaussian orbital based matri elements and similarly for the y- and z-components. The function F ν (η(p Q) ) is evaluated as in (9). 5. Summary and conclusions In a general electron structure calculation, matri elements between quantum state orbitals have to be calculated. We have presented a detailed derivation of the fundamental matri elements in this type of calculations, namely the overlap, kinetic, nuclear attraction and electron repulsion integrals for Gaussian type orbitals. One approach is to use the Gaussian product theorem before epanding the polynomials in Hermite polynomials, and then take the derivatives with respect to atomic positions in order to get the result presented in the article of Clementi and Davis. We have presented a different approach in this article. By epanding a polynomial in Hermite polynomials and using Rodrigues formula, an arbitrary Gaussian type orbital can be epressed as a sum over derivatives of Gaussian functions. The derivatives with respect to atomic positions are taken in order to integrate the Gaussian functions before the derivatives are taken. The derived recursion formulas are not restricted to electron structure calculations; they can be used in, e.g., electrostatistics when the charge density is epanded in terms of Cartesian Gaussian-type functions. References [] Boys S F 95 Proc. R. Soc. A [] Levine I N 999 Quantum Chemistry Fifth Edition (Upper Saddle River: Prentice Hall) [3] Szabo A and Ostlund N S 98 Modern Quantum Chemistry (New York: Macmillan Publishing Company) [4] Thijssen J M 999 Computational Physics (Cambridge: Cambridge University Press) [5] Clementi E and Davis D R 966 J. Comp. Phys. 3 44

Molecular Integral Evaluation

Molecular Integral Evaluation Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 13th Sostrup Summer School Quantum Chemistry and