Fourier Transformation Method for Computing NMR Integrals over Exponential Type Functions

Experimental methods based on magnetic resonance are among the most used techniques for investigating molecular and electronic structure. Nuclear magnetic resonance (NMR) is mostly applied to closed-shell molecules and can be used for structural research of matter in solid, liquid and gaseous form. The computation of NMR parameters, which are of a great interest not only in chemistry but also in biology and solid-state physics, presents severe analytical and numerical difficulties [Dickson & Ziegler (1996); Ditchfield (1974); Fukui et al. (2004); Helgaker et al. (1999); Ishida (2003); London (1937); Pople et al. (1968); Pyykkö (1988); Schreckenbach & Ziegler (1995); Vaara (2007); Watson et al. (2004)]. The computation of NMR parameters for any of the standard models of quantum chemistry constitute an important challenge [Helgaker et al. (1999)]. Calculations involving a magnetic field should preserve gauge invariance. This is conveniently accomplished by using a gauge including atomic orbitals (GIAO) [London (1937)], which is based on atom-centered basis functions with an explicit field dependence. Magnetic properties are sensitive to the quality of the basis sets due to many contributing physical phenomena arising from both the vicinity of the nucleus and from the valence region. A good atomic orbital basis should decay exponentially for large distances [Agmon (1985)] and should also satisfy Kato’s conditions for exact solutions of the appropriate Schrödinger equation [Kato (1957)]. Exponential type functions (ETFs) are better suited than Gaussian functions (GTFs) [Boys (1950a;b)] to represent electron wave functions near the nucleus and at long range. Among the ETFs, Slater type functions (STFs) [Slater (1932)], have a dominating position due to their simple analytic expression, but their multi-center integrals are extremely difficult to evaluate for polyatomic molecules, particularly bi-electronic terms. We note that many researchers hope that the next generation of ab initio programs will be based on the usage of ETFs. Indeed much effort is being made to develop efficient molecular algorithms for integrals over conventional ETFs (STFs or B functions) [Barnett (1990); Fernández et al. (2001); Kutzelnigg (1988); Niehaus et al. (2008); Ozdogan & (Editors); Rico et al. (1998; 1999; 2001); Steinborn et al. (2000); Weatherford & Jones (1982)].


Introduction
Experimental methods based on magnetic resonance are among the most used techniques for investigating molecular and electronic structure.Nuclear magnetic resonance (NMR) is mostly applied to closed-shell molecules and can be used for structural research of matter in solid, liquid and gaseous form.The computation of NMR parameters, which are of a great interest not only in chemistry but also in biology and solid-state physics, presents severe analytical and numerical difficulties [Dickson & Ziegler (1996); Ditchfield (1974); Fukui et al. (2004); Helgaker et al. (1999); Ishida (2003); London (1937); Pople et al. (1968); Pyykkö (1988); Schreckenbach & Ziegler (1995); Vaara (2007); Watson et al. (2004)].The computation of NMR parameters for any of the standard models of quantum chemistry constitute an important challenge [Helgaker et al. (1999)].Calculations involving a magnetic field should preserve gauge invariance.This is conveniently accomplished by using a gauge including atomic orbitals (GIAO) [London (1937)], which is based on atom-centered basis functions with an explicit field dependence.Magnetic properties are sensitive to the quality of the basis sets due to many contributing physical phenomena arising from both the vicinity of the nucleus and from the valence region.A good atomic orbital basis should decay exponentially for large distances [Agmon (1985)] and should also satisfy Kato's conditions for exact solutions of the appropriate Schrödinger equation [Kato (1957)].Exponential type functions (ETFs) are better suited than Gaussian functions (GTFs) [Boys (1950a;b)] to represent electron wave functions near the nucleus and at long range.Among the ETFs, Slater type functions (STFs) [Slater (1932)], have a dominating position due to their simple analytic expression, but their multi-center integrals are extremely difficult to evaluate for polyatomic molecules, particularly bi-electronic terms.We note that many researchers hope that the next generation of ab initio programs will be based on the usage of ETFs.Indeed much effort is being made to develop efficient molecular algorithms for integrals over conventional ETFs (STFs or B functions) [ Barnett (1990); Fernández et al. (2001); Kutzelnigg (1988); Niehaus et al. (2008); Ozdogan & (Editors); Rico et al. (1998;1999;2001); Steinborn et al. (2000); Weatherford & Jones (1982)].
Various studies focussed on the use of B functions.The use of B functions was proposed by Shavitt [Shavitt (1963)], since reduced Bessel functions possess a representation in terms of a remarkably simple Gauss transform.Detailed discussions of the mathematical properties of reduced Bessel functions and of their anisotropic generalizations can be found in [Weniger (1982)].Furthermore, B functions have much more appealing properties applicable to multi-center integral problems, compared to other exponentially decaying functions [Filter & Steinborn (1978a;b); Steinborn & Filter (1975); Weniger (2005); Weniger & Steinborn (1983a)].The multi-center molecular integrals over B functions can be computed much more easily than the corresponding integrals of other exponentially decaying functions.This can be explained in terms of the Fourier transform of B functions, which is of exceptional simplicity among exponentially decaying functions [Niukkanen (1984); Weniger (1982); Weniger & Steinborn (1983b)].Moreover, the Fourier transforms of STFs, of hydrogen eigenfunctions, or of other functions based on the generalized Laguerre polynomials can all be expressed as finite linear combinations of Fourier transforms of B functions [Weniger (1985); Weniger & Steinborn (1983b)].The basis set of B functions is well adapted to the Fourier transform method [Geller (1962); Grotendorst & Steinborn (1988); Prosser & Blanchard (1962); Trivedi & Steinborn (1983)], which allowed analytic expressions to be developed for molecular multi-center integrals over B functions [Grotendorst & Steinborn (1988); Trivedi & Steinborn (1983)].
Of the NMR parameters, the nuclear shielding tensor is of a great importance.The computation of the shielding tensor presents severe analytical and numerical difficulties especially when using ETFs as a basis set of atomic orbitals.The main difficulty arises from the operators associated with these parameters.An example of such operators is 3 r jN,β r jN • σ(j) /r 5 jN , where β represents a cartesian coordinates, r jN is the vector separating the j th electron and the N th nuclei and σ stands for Pauli spin matrix.These operators lead to extremely complicated integrals.Analytic treatment of the NMR parameters over GTFs was a subject of many articles (see for example [Ishida (2003)] and references therein).Although, the interest of using ETFs in the computation of NMR parameters is increasing [see the pioneer work by Dickson & Ziegler (1996); Schreckenbach & Ziegler (1995)] and Watson et al. (2004)], no effort was dedicated to their analytic treatment over ETFs.Straightforward numerical integration was used for the computation of integrals associated with these parameters.

Molecular integrals in the absence of magnetic fields
In the absence of magnetic fields, the molecular electronic Hamiltonian operator H e corresponding to total energy E e for a molecule of N nuclei and n e electron is given by: where: • Z K is the atomic number of the K th nucleus whose mass is M K .
• r jK is the distance that separates the K th nucleus from the j th electron.
• r ij is the distance that separates the i th electron from the j th electron.
• ∇ 2 j is the Laplacian operators for the coordinates of electron j.
The stationary Schrödinger equation that needs to be solved is: The above Schrödinger equation is solved only in the case of hydrogen-like atom.The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals.These atomic orbitals form a complete and orthonormal basis.The use of hydrogen-like atomic orbitals was prevented due to the fact that their molecular multi-center integrals are extremely difficult to evaluate analytically and numerically.Linear combinations of the hydrogen-like atomic orbitals lead to STFs, which form the most popular basis set of atomic orbitals.Unormalized STFs are given by [Slater (1932)]: where Y m l (θ, ϕ) is the surface spherical harmonic [Condon & Shortley (1951)] and where n is the principal quantum number, l is the orbital angular momentum number and m is the magnetic quantum number.

Relativistic formulation of NMR shielding tensor
In the presence of an external uniform magnetic field B 0 , the electronic non-relativistic Hamiltonian is given by: where the electronic impulsion p j is given by: where A i stands for the vector potential induced by the nuclear moments μ N and the external uniform magnetic field B 0 .μ 0 stands for dielectric permittivity.r ij is the modulus of the vector r ij separating the electrons i and j. r iN is the vector separating the electron i and the nuclei N.
The relativistic effects are important for the fourth and fifth rows in the periodic table and for transitions metals [Pyykkö (1988)].In terms of perturbations with respect to μ N,α and B 0,β where α and β stand for cartesian coordinates (α, β ∈ (x, y, z)), the electronic relativistic Hamiltonian is given by: 125 Fourier Transformation Method for Computing NMR Integrals over Exponential Type Functions www.intechopen.comwhere H (0) is the zeroth-order hamiltonian (1) and H (r) is the relativistic perturbation term, which is independent of the magnetic perturbations and is given by [Fukui & Baba (1998)]: which include the contributions; H (r) 1,j : two-electron Darwin term, H 2,j : two-electron spin-orbit term, H (r) 3,j : the retarded orbit-orbit term, H (r) 4,j : spin-other-orbit term and H (r) 5,j : spin-spin term.
The perturbations are given by [Fukui & Baba (1998)]: H (1,1) where l jX = −i r jX ∧ ∇ j , l j = −i r j ∧ ∇ j and σ j stands for the Pauli spin matrix of the electron j and its cartesian coordinates are given by: σ j,x = 01 10 , σ j,y = 0 i i 0 and σ j,z = 10 0 −1 .

Fourier transformation for the analytic development of NMR integrals
The operators involved in equations ( 18), ( 19) and ( 20) lead to very complicated integrals.
The analytic development of the these NMR integrals is difficult due to the presence of the operator involving 1/r n , which is not the case of the usual three-center molecular integrals (zeroth order molecular integrals) where the Coulomb operator 1/r is involved.
In this review, we present the method based on Fourier transform that led to analytic expressions for first and second order integrals of the shielding tensor.Among the operators involved in integrals of the shielding tensors are r j • r jN δ αβ − r jN,α r j,β r 3 jN in equation ( 20) for the second order terms and 3r jN,β r jN • σ j r 5 jN in equation ( 18) in the case of first order relativistic terms.
The integrals induced by the above operators are given by: 126 Fourier Transform -Materials Analysis www.intechopen.com where r jA = r j − # -OA, r jB = r j − # -OB and r jN = r j − # -ON.A, B and N are three arbitrary points of the Euclidean space and 0 is the origin of the fixed coordinate system.

First order integrals
After expanding the operator in the integrals (21), we can write I 13 as follows: Let I (α,β) 13 represent the integrals in summation of the RHS of the above equation.Using equation ( 5), the integrals I (α,β) 13 are expressed as linear combinations of integrals B I (α,β) 13 over B functions which are given by: Using the Fourier transform method, we obtain: (25) In the case where α and β represent two different cartesian coordinates, the Fourier transform of the operator is given by [Safouhi (2010b)]: In the case where α and β represent the same cartesian coordinate, the calculations leads to the potential 1/r 3 , which poses serious difficulties because of the singularity and its Fourier transform does not exist in a sense of classical analysis.This case is a part of ongoing research where the theory of generalized functions will be used in order to derive the Fourier transform of the operator in the generalized function sense.
Using the analytic expression (11) obtained by Trivedi and Steinborn [Trivedi & Steinborn (1983)] for the integrals over r j involved in equation ( 25) and with the help of equation ( 26), one can derive the following analytic expression for the integrals B I (α,β) 13 [Safouhi (2010b)]: Fourier Transformation Method for Computing NMR Integrals over Exponential Type Functions where: In the case of one-and two-center integrals B I (α,β) 11 and B I (α,β) 12 , corresponding to A = B = N and A = B = N respectively, we derived analytical expressions in [Slevinsky et al. (2010)].

For the one-center integrals (
where: For the two-center integrals (A = B = N):

Second order integrals
The second order integrals I 23 are given by ( 22).For simplicity and without loss of generality, we assume A = O in equation ( 25).
The operator involved in ( 22) is given by: From the above equation, it is obvious that the integrals I 23 can be expressed in terms of integrals of the form: 129 Fourier Transformation Method for Computing NMR Integrals over Exponential Type Functions www.intechopen.com The cartesian coordinate r jN,α and r j,β can be expressed in terms of spherical harmonics and r jN and r j respectively as follows: where the coefficients c α,μ are given as follows: Using the analytic expression of the Unormalized STFs (2), one can obtain: The product of two spherical harmonics can be linearized by Gaunt coefficients as follows: where the summation index l runs in steps of 2 from min to l 1 + l 2 .The constant l min is given by [Weniger & Steinborn (1982)]: From equation (37), it follows that: Using equations ( 34) and (39), we obtain the following relation: where m1 = m 1 − μ 2 and ñ1 = n 1 + 1.
Using the Fourier transform method, we obtain: The Fourier transform of the operator involved in the above integrals is given by [Berlu & Safouhi (2008)]: Using the analytic expression (11) obtained by Trivedi and Steinborn [Trivedi & Steinborn (1983)] for the integrals over r j involved in equation ( 42) and equation ( 43), one can derive the following analytic expression for the integrals B Îαβ 23 [Berlu & Safouhi (2008)]:   29).

Conclusion
In the present review, we showed how the Fourier transformation method allowed the derivation of compact formulae for one of the most challenging integrals, namely molecular multi-center integrals and NMR multi-center integrals.
Analytic expressions are obtained for integrals of the paramagnetic contribution in the relativistic calculation of the shielding tensor as well as integrals of second order in the non-relativistic calculation of the shielding tensor.The basis set of ETFs is used and it is well known that these functions are better suited than GTFs.
The obtained analytic expressions for the one-and two-center integrals can be computed easily and no quadrature rule is required.In the case of the three-center integrals, we need to compute semi-infinite integrals involving oscillatory functions.These oscillatory integrals can be computed to a high pre-determined accuracy using existing methods and algorithms based on extrapolation methods and numerical quadrature [Berlu & Safouhi (2003); Duret & Safouhi (2007);Safouhi (2001b;2004;2010a); Slevinsky & Safouhi (2009)].

Acknowledgment
The author acknowledges the financial support for this research by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Table 1 .
Evaluation of B I