Explicitly correlated Gaussians with tensor pre-factors: analytic matrix elements

We consider a specific form of explicitly correlated Gaussians—with tensor pre-factors—which appear naturally when dealing with certain few-body systems in nuclear and particle physics. We derive analytic matrix elements with these Gaussians—overlap, kinetic energy, and Coulomb potential—to be used in variational calculations of those systems. We also perform a quick test of the derived formulae by applying them to p-and d-waves of the hydrogen atom


Introduction
Explicitly correlated Gaussians in various forms is a popular basis for variational calculations of quantummechanical few-body systems [1,2,3].One important advantage of Gaussians is that their matrix elements are often analytic [2,4,5,6,7] which greatly facilitates the calculations.
The most general Gaussians are shifted correlated Gaussians.However, they have a disadvantage: they lack a definite angular momentum.That might be problematic in applications where angular momentum is an important quantum number.In the literature one can find analytic formulae with Gaussians that acquire a definite angular momentum via pre-factors built of spherical harmonics [6].However there exist applications in nuclear and particle physics, in particular with pions [8,9,10], where the variational basis-functions are multiplied by operators in the form of vectors and tensors 1 .These applications favour Gaussians with vector and tensor pre-factors rather than spherical harmonics.In this contribution we derive the analytic matrix elements with such Gaussians.We start with the matrix elements of the shifted correlated Gaussians, perform their Taylor expansion with respect to shift-vectors, and then collect the terms of specific orders, which ultimately gives the sought matrix elements.
Since some of the resulting formulae are rather long, particularly for the tensor pre-factor, we perform a quick test of the derived formulae by applying them to the lowest p-and d-wave states of the hydrogen atom.

Shifted Gaussians
Since we are going to use-as the starting point of our derivations-the analytic matrix elements with shifted correlated Gaussians we reproduce here the relevant analytic formulae from [7].
For an n-body system of particles the shifted correlated Gaussian with correlation matrix A and shift a is defined in coordinate space r as where T denotes transposition, r is the column {⃗ r 1 , . . ., ⃗ r n } of the coordinates of the bodies, the shift a is a column of shift vectors {⃗ a 1 , . . ., ⃗ a n }, and where where the central dot denotes the scalar product of two vectors.The overlap of two shifted Gaussians is given as ⟨B, b|A, a⟩ = e 1 4 (a+b) T R(a+b) where R = (A + B) −1 .
The matrix element of the kinetic energy operator is given as where K is the (reduced) mass-matrix of the system of particles in the given set of coordinates.
The matrix element of the Coulomb potential is given as where w is a given column of numbers {w 1 , . . ., w n }, and where β .
3 Rank-0 Gaussians The matrix elements with rank-0 (s-wave) Gaussians are well known.Notwithstanding, we reproduce here the relevant formulae for completeness.A rank-0 Gaussian is the zero-shift limit of a shifted Gaussian, The overlap of two rank-0 Gaussians is the zero-shift limit of the shifted overlap (4), Kinetic energy matrix element with rank-0 Gaussians is again given as the zero-shift limit of the corresponding shifted matrix element (5), The Coulomb potential matrix element is, analogously, the zero-shift limit of the shifted matrix element (6), 4 Rank-1 pre-factor Gaussians The rank-1 pre-factor Gaussians are constructed by pre-factoring rank-0 Gaussians with the form (a T r) (which represents a pure p-wave).We shall use the following notation, where a is a column of polarization vectors {⃗ a 1 , . . ., ⃗ a n }, and |(a)A⟩ is our rank-1 Gaussian.
The rank-1 Gaussian can be obtained by collecting the linear term in the Taylor expansion of the shifted Gaussian in the shift-variable, ⟨r|A, a⟩ .= e −r T Ar+a T r = e −r T Ar + (a T r)e −r T Ar + 1 2 (a T r) 2 e −r T Ar + . . . .
We shall denote this by the following notation, The overlap of rank-1 Gaussians can be obtained by collecting the terms on the order O(ab) from the Taylor expansions of the shifted overlap (and employing the symmetry of the matrix R), The O(ab) term of the expansion is given as Similarly, the kinetic energy matrix element with rank-1 Gaussians can be obtained by collecting the terms on the order O(ab) from the Taylor expansion of the corresponding shifted Gaussian matrix element, The Taylor expansion of the shifted kinetic energy matrix element (5) gives, term by term, Summing up, The Coulomb matrix element is again obtained by collecting the terms O(ab) from the Taylor expansion of the shifted Coulomb matrix element (6), We first expand the error-function, 2 ⟨B, b| where Now, performing full expansion and collecting the terms O(ab), gives 5 Rank-2 pre-factor Gaussians The rank-2 pre-factor Gaussians are constructed as where a and b are the polarization vectors.The rank-2 pre-factor generally contains both s-waves and d-waves, however the condition a T b = 0 eliminates the s-wave contribution.The rank-2 Gaussian is the O(ab) term in the Taylor expansion of the shifted Gaussian, The overlap of rank-2 Gaussians is given by the terms O(abcd) from the Taylor expansion of the shifted overlap, Performing the Taylor expansion, gives

Kinetic energy
The rank-2 kinetic energy matrix element is given by the sum of the O(abcd) terms in the Taylor expansion of the shifted Gaussian kinetic energy (5).Performing the Taylor expansion of (5) gives the following, term by term,

Coulomb
The rank-2 Coulomb matrix element is given by the sum of the terms O(abcd) in the Taylor expansion of the shifted Gaussian Coulomb matrix element (6), where Performing the expansion and collecting the O(abcd) terms gives The rank-2 Gaussians are taken in the form where the polarization vectors are chosen as ⃗ a = {1, 0, 0}, ⃗ b = {0, 1, 0} such that ⃗ a • ⃗ b = 0 and the Gaussian represents a pure d-wave, The Schrodinger equation in the space spanned by the given set of Gaussians is represented by the generalized matrix eigenvalue problem, where N (l) is the overlap matrix, N H (l) is the Hamiltonian matrix, H and n } is the column of the expansion coefficients.The matrix elements are calculated using the derived formulae.
The generalized matrix eigenvalue problem is solved using the standard method (via Cholesky decomposition of the overlap matrix).The parameters α i of the Gaussians are tuned to minimize the lowest eigenvalue by a gradient-descent method.The results of the calculations are shown on figure (7) where the lowest energies for s-, p-, and d-waves are shown as functions of the number of Guassians in the variational basis.The exact energies3 are reproduced within 4 decimal digits with the basis of 5 Gaussians.

Conclusion
We have derived analytic matrix elements-overlap, kinetic energy, and Coulomb potential-for correlated Gaussians with tensor pre-factors.Tensor pre-factor Gaussians might be useful in certain nuclear and particle physics applications, in particular when pions are included explicitly.We have done a quick test of the derived formulae by applying them to the p-and d-waves of the hydrogen atom.