Evaluation of Gaussian Molecular Integrals

Transcription

1 The Mathematica Journal Evaluation of Gaussian Molecular Integrals II. Kinetic-Energy Integrals Minhhuy Hô Julio Manuel Hernández-Pérez This article carries out the evaluation of kinetic energy integrals using Gaussian-type functions with arbitrary Cartesian angular values. As an example, we calculate the kinetic matrix for the water molecule in the STO-3G basis set. Introduction In this article, the second of a series describing algorithms for evaluating molecular integrals, we detail the evaluation of the kinetic energy integrals. Detailed accounts of molecular integrals can be found in the references of [1]. The electronic kinetic energy in atomic units (m e = e = = 1 ê 4 p e 0 = 1) involves integrals of the type K = - c a Hr; a, A, al c b Hr; b, B, bl dr, (1) in which c a HrL is an unnormalized Cartesian Gaussian primitive centered at the nucleus A = 9A x, A y, A z =: c a Hr; a, A, al = Hx - A x L a x Iy - A y M a y Hz - A z L a z e -a r-a 2 = Hx - A x L a x e -a x-a x 2 Iy - A y M a y e -a y-a y 2 Hz - A z L a z e -a z-a z 2 = c a Hx; a, A x, a x L ÿ c a Iy; a, A y, a y M ÿ c a Hz; a, A z, a z L. Here a is the orbital exponent and the polynomial represents the angular part, in that the sum of the Cartesian angular momenta a x + a y + a z = 0, 1, 2, 3, corresponds to functions of type s, p, d, f,. One notable property of the Gaussian function, which will be used here, is that the derivative of a Gaussian function can be expressed as a sum containing Gaussians of lower and higher Cartesian angular values. In particular, x c ahx; a, A x, a x L = a x c a Hx; a, A x, a x - 1L - 2 a c a Hx; a, A x, a x + 1L. (2) (3)

2 2 Minhhuy Hô and Julio Manuel Hernández-Pérez Similarly, A x c a Hx; a, A x, a x L = - x c ahx; a, A x, a x L. This property is useful in integrals involving differential operators, such as the kinetic energy; in calculations involving the gradient of the energy or Hamiltonian; and in deriving one of the most important algorithms of Gaussian function computation, the recurrence relation. The kinetic operator, expressed in Cartesian coordinates as = J 2 x y z 2 N, enables separation of the kinetic integrals into three components K = K x + K y + K z. For example, K x = c a Hr; a, A, al 2 x 2 c bhr; b, B, bl dr. Furthermore, if we also separate c a into its Cartesian components, K x becomes a product of three integrals K x = c a Hx; a, A x, a x L - (4) 2 x c bhx; b, B x, b x L dx c a Iy; a, A y, a y M ÿ c b Iy; b, B y, b y M 2 - (5) dy - c a Hz; a, A z, a z L ÿ c b Hz; b, B z, b z L dz. Derivative of Gaussian Functions The last two integrals in K x are simply overlap integrals and can be evaluated as outlined in [1]. Integrating by parts, the first integral is Bc ahx; a, A x, a x L - - x= x c bhx; b, B x, b x L x=- x c ahx; a, A x, a x L x c bhx, b, B x, b x L dxf = x c ahx; a, A x, a x L x c bhx, b, B x, b x L dx where, according to the property mentioned above, the first term can be expanded into two Gaussian products, which vanish at the integral limits. (6)

3 Evaluation of Gaussian Molecular Integrals 3 Next, we consider the first term of the integrand. Expressing equation (3) in terms of (1) and rearranging, we get x Hx - A xl a x e -ahx-a xl2 = a x Hx - A x L a x-1 e -ahx-a xl 2 ahx - A x LD Hx - A x L a x e -ahx-a xl 2 = A a x Hx - A x L a x-1-2 ahx - A xl a x+1 E e -ahx-a xl 2. Substituting the result into equation (6), we have 1 2 Aa x Hx - A x L ax-1-2 ahx - A xl ax+1 E - e -ahx-a xl 2 Ab x Hx - B x L b x-1-2 bhx - B xl b x+1 E e -bhx-b xl 2 dx, and expanding the polynomial part yields 1 2 Aa x b x Hx - A x L ax-1 Hx - B x L bx a b x Hx - A xl a x+1 Hx - B x L b x-1-2 a x bhx - A x L a x-1 Hx - B x L b x a bhx - A x L a x+1 Hx - B x L b x+1 E e -ahx-a xl 2 e -bhx-b xl 2 dx. We can now express the first integral of equation (5) in terms of one-dimensional Gaussian primitives: 1 2 Ba x b x - c a Hx; a, A x, a x - 1L ÿ c b Hx; b, B x, b x - 1L dx - (7) (8) (9) 2 a b x - c a Hx; a, A x, a x + 1L ÿ c b Hx; b, B x, b x - 1L dx - 2 a x b - c a Hx; a, A x, a x - 1L ÿ c b Hx; b, B x, b x + 1L dx + (10) 4 a b - c a Hx; a, A x, a x + 1L ÿ c b Hx; b, B x, b x + 1L dxf. Kinetic-Energy Integral We have now shown that the kinetic-energy integral can be written as a product of overlap integrals. The x component, for example, is given by K x = 1 2 Ba x b x - c a Hx; a, A x, a x - 1L ÿ c b Hx; b, B x, b x - 1L dx - 2 a b x - c a Hx; a, A x, a x + 1L ÿ c b Hx; b, B x, b x - 1L dx - 2 a x b - c a Hx; a, A x, a x - 1L ÿ c b Hx; b, B x, b x + 1L dx + (11)

4 4 Minhhuy Hô and Julio Manuel Hernández-Pérez 4 a b - c a Hx; a, A x, a x + 1L ÿ c b Hx; b, B x, b x + 1L dxf - c a Iy; a, A y, a y M ÿ c b Iy; b, B y, b y M dy - c a Hz; a, A z, a z L ÿ c b Hz; b, B z, b z L dz. Recurrence Relations Using the notation of [1], where the overlap integral of two Gaussians is expressed in terms of its orthogonal components, S = c a Hx; a, A x, a x L ÿ c b Hx; b, B x, b x L dx = - p E AB B F3ê2 sx Ha x, b x L s y Ia y, b y M s z Ha z, b z L, the x component of the kinetic-energy integral, equation (11), can be written as K x = 1 2 E p ABB F3ê2 8ax b x s x Ha x - 1, b x - 1L - 2 a b x s x Ha x + 1, b x - 1L - 2 a x bs x Ha x - 1, b x + 1L + 4 a b s x Ha x + 1, b x + 1L< s y Ia y, b y M p s z Ha z, b z L = E AB B F3ê2 kx Ha x, b x L s y Ia y, b y M s z Ha z, b z L, (12) (13) where the factor 1 2 has been absorbed into the definition of the kinetic integral k xha x, b x L. We will derive the recurrence relation for this term k x Ha x, b x L. For a x = b x = 0, the first three terms of equation (13) are zero, and we are left with k x H0, 0L = 2 a b s x H1, 1L. Analogously, for a x = 0 or b x = 0, we have k x Ha x, 0L = -a x b s x Ha x - 1, 1L + 2 a b s x Ha x + 1, 1L, k x H0, b x L = -a b x s x H1, b x - 1L + 2 a b s x H1, b x + 1L, and the recurrence relation for the kinetic integral function with any two Cartesian angular momenta generalizes to (14) k x Ha x, b x L = 1 x b x s x Ha x - 1, b x - 1L - 2 a b x s x Ha x + 1, b x - 1L - 2 a x b s x Ha x - 1, b x + 1L + 4 a b s x Ha x + 1, b x + 1LD. (15)

5 Evaluation of Gaussian Molecular Integrals 5 Implementation The function Kin evaluates the kinetic integral of two Gaussian primitives; here alpha, beta, RA, RB, LA, and LB are a, b, A, B, a, and b as defined earlier. The first step is the evaluation of the overlap integral as described in [1]. Kin@alpha_, beta_, RA_, RB_, LA_, LB_D := Module@8EAB, Kinetic<, Do@ H* Initial Conditions for overlap recurrence relation *L s@i, 0, 0D = 1; s@i, 1, 0D = -HRA@@iDD - HHalpha * RA@@iDD + beta * RB@@iDDL ê HalphetaLLL; H* Recurrence Index *L s@i_, a_, 0D := -HRA@@iDD - Halpha * RA@@iDD + beta * RB@@iDDL ê HalphetaLL * s@i, a - 1, 0D + HHa - 1L ê H2 * HalphetaLLL * s@i, a - 2, 0D; H* Transfer Equation *L s@i_, a_, b_d := s@i, a + 1, b - 1D + HRA@@iDD - RB@@iDDL * s@i, a, b - 1D; H* Initial Conditions for kinetic integral function *L k@i, 0, 0D = 2 * alpha * beta * s@i, 1, 1D; k@i_, a_, 0D := -a * beta * s@i, a - 1, 1D + 2 * alpha * beta * s@i, a + 1, 1D; k@i_, 0, b_d := -alpha * b * s@i, 1, b - 1D + 2 * alpha * beta * s@i, 1, b + 1D; H* Kinetic Energy Integral *L k@i_, a_, b_d := Ha * b * s@i, a - 1, b - 1D - 2 * a * beta * s@i, a - 1, b + 1D - 2 * alpha * b * s@i, a + 1, b - 1D + 4 * alpha * beta * s@i, a + 1, b + 1DL ê 2;, 8i, 1, 3<D; H* Loop over three Cartesian components *L H* Exponential Overlap Factor *L

6 6 Minhhuy Hô and Julio Manuel Hernández-Pérez EAB = Exp@-Halpha * beta ê HalphetaLL * HRA - RBL.HRA - RBLD; H* Kinetic Energy Integral *L D Kinetic = EAB * HHPi ê HalphetaLL^H3 ê 2LL * Hk@1, LA@@1DD, LB@@1DDD * s@2, LA@@2DD, LB@@2DDD * s@3, LA@@3DD, LB@@3DDD + s@1, LA@@1DD, LB@@1DDD * k@2, LA@@2DD, LB@@2DDD * s@3, LA@@3DD, LB@@3DDD + s@1, LA@@1DD, LB@@1DDD * s@2, LA@@2DD, LB@@2DDD * k@3, LA@@3DD, LB@@3DDDL We describe in detail the evaluation of the kinetic-energy matrix for the water molecule (r OH = bohr, HOH = , the geometry optimized at the HF/STO-3G level). The molecule lies in the y-z plane with Cartesian coordinates in atomic units. R = 880., , <, 80., , <, 80., 0., <<; In the STO-3G basis set, each atomic orbital is approximated by a sum of three Gaussians; their unnormalized primitive contraction coefficients and orbital exponents (taken from [2]) are as follows. PrimCoeff = , , <, , , <, , , <, , , <, , , <, , , <, , , <<; OrbCoeff = , , <, , , <, , , <, , , <, , , <, , , <, , , <<;

7 Evaluation of Gaussian Molecular Integrals 7 Here are basis function origins and Cartesian angular values of the orbitals, listed in the order H 1 H1 sl, H 2 H1 sl, O 3 H1 sl, O 3 H2 sl, O 3 H2 p x L, O 3 I2 p y M, and O 3 H2 p z L. FCenter = 8R@@1DD, R@@2DD, R@@3DD, R@@3DD, R@@3DD, R@@3DD, R@@3DD<; CartAng = 880, 0, 0<, 80, 0, 0<, 80, 0, 0<, 80, 0, 0<, 81, 0, 0<, 80, 1, 0<, 80, 0, 1<<; Specifically, for the kinetic-energy integral of the first primitive of the 1 s orbital of hydrogen atom 1, c 11 HrL, and the first primitive of the 2 p z orbital of the oxygen atom, c 71 HrL, - c 11 HrL c 71 HrL dr = e r -R H Hz L µ (16) e r -R O3 2 dx dy dz, where the indices c ij indicate primitive j of basis function i. The kinetic integral in terms of the kinetic and overlap function is then p E AB B F3ê2 9k x H0, 0L s y H0, 0L s z H0, 1L + s x H0, 0L k y H0, 0L s z H0, 1L + s x H0, 0L s y H0, 0L k z H0, 1L=. We start with a = 80, 0, 0< and b = 80, 0, 1<, and from the first four equations of the module, we use the recurrence scheme to build up the overlap values needed for k z H0, 1L. We will need values up to s z H1, 2L, s z H0, 0L = 1, (17) s z H1, 0L = - H1 z - a H1 z + b O z s z H2, 0L = - H1 z - a H1 z + b O z s z H3, 0L = - H1 z - a H1 z + b O z = , s z H1, 0L HL s zh0, 0L = , s z H1, 1L = s z H2, 0L + HH1 z - O z L s z H1, 0L = , s z H2, 1L = s z H3, 0L + HH1 z - O z L s z H2, 0L = , s z H1, 2L = s z H2, 1L + HH1 z - O z L s z H1, 1L = , s z H2, 0L HL s zh1, 0L = , (18) leading to k z H0, 1L = -a ÿ 1 ÿ s z H1, 0L + 2 a ÿ b ÿ s z H1, 2L =

8 8 Minhhuy Hô and Julio Manuel Hernández-Pérez Similarly, for k x H0, 0L, we will need s x H1, 1L which, in turn, comes from s x H1, 0L and s x H2, 0L. We have s x H1, 0L = 0, s x H2, 0L = , s x H1, 1L = , leading to k x H0, 0L = 2 a b s x H1, 1L = For the y component, s y H1, 0L = , s y H2, 0L = , s y H1, 1L = , leading to k y H0, 0L = With E AB = , the kinetic integral of equation (16) is K = , which we can also obtain from the module Kin. Kin@OrbCoeff@@1, 1DD, OrbCoeff@@7, 1DD, FCenter@@1DD, FCenter@@7DD, CartAng@@1DD, CartAng@@7DDD Three Gaussian primitives c a HrL for each atomic orbital f i HrL result in nine integrals of the type that we have just evaluated. For example, the K 17 element of the kinetic-energy matrix, 3 3 K 17 = N 1 i N 7 j d 1 i d 7 j c 1 i HrL c 7 j HrL dr, i=1 j=1 (19) is Ÿ f1 - HrL J N f 7 HrL dr, derived via a contraction scheme that requires the following normalization factor. NormCoeff@a_, 8ax_, ay_, az_<d := 2 a p 3ê4 H4 al Hax+ay+azLê2 ; HH2 ax - 1L!! H2 ay - 1L!! H2 az - 1L!!L 1ê2

9 Evaluation of Gaussian Molecular Integrals 9 This calculates K 17. Sum@ NormCoeff@OrbCoeff@@1, idd, CartAng@@1DDD * NormCoeff@OrbCoeff@@7, jdd, CartAng@@7DDD * PrimCoeff@@1, idd * PrimCoeff@@7, jdd * Kin@OrbCoeff@@1, idd, OrbCoeff@@7, jdd, FCenter@@1DD, FCenter@@7DD, CartAng@@1DD, CartAng@@7DDD, 8i, 3<, 8j, 3<D For larger basis sets, one needs only to replace the summation upper limit 3 in equation (19) with the appropriate number of primitives belonging to a particular basis function and an additional summation for the basis functions. Finally, here is the resulting kinetic energy matrix. Table@Sum@ NormCoeff@OrbCoeff@@p, idd, CartAng@@pDDD * NormCoeff@OrbCoeff@@q, jdd, CartAng@@qDDD * PrimCoeff@@p, idd * PrimCoeff@@q, jdd * Kin@OrbCoeff@@p, idd, OrbCoeff@@q, jdd, FCenter@@pDD, FCenter@@qDD, CartAng@@pDD, CartAng@@qDDD, 8i, 3<, 8j, 3<D, 8p, 1, 7<, 8q, 1, 7<D êê MatrixForm êê Chop Since the kinetic-energy matrix is symmetrical, we need only to calculate the upper elements. The value of the elements K 11 = K 22 = hartree is the electronic kinetic energy of the hydrogen atom described by the STO-3G basis set (compared to the exact value of 0.5 hartree). Analogously, the element K 33 is the corresponding kinetic energy of an electron in the 1 s orbital of the oxygen atom, and so forth.

10 10 Minhhuy Hô and Julio Manuel Hernández-Pérez Conclusion We have provided an introduction to the evaluation of kinetic-energy integrals involving Gaussian-type basis functions both analytically and by use of recurrence relations. The results are sufficiently general so that no modification of the algorithm is needed when larger basis sets with more Gaussian primitives or primitives with larger angular momenta are employed. References [1] M. Hô and J. M. Hernández-Pérez, Evaluation of Gaussian Molecular Integrals I, The Mathematica Journal, doi: /tmj [2] Basis Set Exchange. (April 12, 2012) bse.pnl.gov/bse/portal. M. Hô and J. M. Hernández-Pérez, Evaluation of Gaussian Molecular Integrals II, The Mathematica Journal, dx.doi.org/doi: /tmj About the Authors Minhhuy Hô received his Ph.D. in theoretical chemistry at Queen s University, Kingston, Ontario, Canada in He is currently a professor at the Centro de Investigaciones Químicas at the Universidad Autónoma del Estado de Morelos in Cuernavaca, Morelos, México. Julio-Manuel Hernández-Pérez obtained his Ph.D. at the Universidad Autónoma del Estado de Morelos in He has been a professor of chemistry at the Facultad de Ciencias Químicas at the Benemérita Universidad Autónoma de Puebla since Minhhuy Hô Universidad Autónoma del Estado de Morelos Centro de Investigaciones Químicas Ave. Universidad, No. 1001, Col. Chamilpa Cuernavaca, Morelos, Mexico CP homh@uaem.mx Julio-Manuel Hernández-Pérez Benemérita Universidad Autónoma de Puebla Facultad de Ciencias Químicas Ciudad Universitaria, Col. San Manuel Puebla, Puebla, Mexico CP julio.hernandez@correo.buap.mx