Integral representations over finite limits for quantum amplitudes

We extend prior work to derive three additional M-1-dimensional integral representations--over the interval $[0,1]$ --for products of M Slater orbitals that allows their magnitudes of coordinate vector differences (square roots of polynomials) $|{\bf x}_{1}-{\bf x}_{2}|=\sqrt{x_{1}^{2}-2x_{1}x_{2}\cos\theta+x_{2}^{2}}$ to be moved from disjoint products of functions into a single quadratic form whose square my be completed. This provides more alternatives to Fourier transforms that introduce a 3M-dimensional momentum integral for those products of Slater orbitals, followed by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current work is also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals. We have found that two of these M-1-dimensional integral representations over the interval $[0,1]$ are numerically stable, as was the prior version having integrals running over the interval $[0,\infty]$, and one does not need to test for a sufficiently large upper integration limit. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over $[0,1]$ than over $[0,\infty]$. These representations have integration variables within square roots as arguments of Macdonald functions. In a number of cases, these may be converted to Meijer G-functions for which a single tabled integral exists over the interval $[0,\infty]$ of the prior paper, and from which other forms may be found. Finally, we introduce a fourth integral representation that is not easily generalizable to large M, but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over $[0,1]$.

A prior paper [11] introduced a fifth reduction method in the spirit of Fourier and Gaussian transforms that is an integral representation having one fewer integral dimension than does a Gaussian transform to represent a products of M Slater functions, and roughly 4M fewer integral dimensions than does a Fourier transform for such a product.This is an advantage since the main drawback of using integral representations is that one adds to the number of integral dimensions one must ultimately solve.In each of these three methods, the reduction of those introduced integrals becomes more difficult the larger the numbers of wave functions transformed, so one-fewer dimension is not a trivial advantage.
Gaussian transforms require a single one-dimensional integral for each wave function, and the completion of the square in the coordinate variables can be done in the resulting exponential.For Fourier transforms, on the other hand, one must introduce a three-dimensional integral for each wave function, and often additional integrals to combine the resulting momentum denominators into a single denominator so that one can complete the square in the momenta to allow the angular integrals to be performed.[12] Our prior work requires the introduction of one fewer integral dimension than does a Gaussian transform set, and many fewer than for Fourier transforms.Its main downside is that the resulting quadratic form (whose square one will complete) resides in a square root as the argument of a Macdonald function, for which there are fewer tabled integrals than for the exponential function wherein resides the quadratic form of Gaussian transforms.
The present paper derives four compact integral representations over finite intervals to represent a product of Slater-type atomic orbitals, the seed function ψ 000 from which Slater functions, [13] Hylleraas powers, [7] and hydrogenic wave functions are derived by differentiation.(Known as the Yukawa [14] exchange potential in nuclear physics, this function also appears in plasma physics, where it is known as the Debye-Hückel potential, arising from screened charges [15] requiring the replacement of the Coulomb potential by an effective screened potential.[16,17] Such screening of charges also appears in solid-state physics, where this function is called the Thomas-Fermi potential.In the atomic physics of negative ions, the radial wave function is given by the equivalent Macdonald function R(r) = C √ r K 1/2 (ηr) .[18] This function also appears in the approximate ground state wave function [19] for a hydrogen atom interacting with hypothesized non-zero-mass photons.[20] We will simply call these Slater orbitals herein.) We start with the simplest integral requiring transformation, the product of two Slater orbitals integrated over all space, where we use the much more general notation of previous work [10] in which the short-hand form for shifted coordinates is x 12 = x 1 − x 2 , p 1 is a momentum variable within any plane wave associated with the (first) integration variable, the y i are coordinates external to the integration, and the j s are defined in the Gaussian transform [10] of the generalized Slater orbital: In our prior work we showed how Gaussian transforms can reduce this integral in roughly eight steps, so this time we will use Fourier Transforms: [21,22,23] ) This is a considerably lengthy derivation, and the Gaussian Transform approach is not much better.Of course, in this simple case, one can invoke the addition theorem expression for e −i k2•x2 in the fourth line to shorten the reduction, but for the large-M equivalent, the above process is what one must follow .(Actually, one would do well to avoid the use of the Dirac delta function in the third line when dealing with large M.) This was the motivation for the fifth path to a solution of our prior paper.

A simpler integral representation
We begin by introducing an integral representation over a finite interval for a pair-product of Slater orbitals, whose derivation will follow a display of its utility.We insert it in the above problem, completing the square in the quadratic form (changing variables from x 2 with unit Jacobian) in both places in the integrand where it appears, so that [24,25] which is indeed a much shorter path to the solution than the Fourier and Gaussian transforms give.This follows from the fact that it requires the introduction of one integral to represent a pair-product of Slater orbitals rather than one integral for each orbital that the Gaussian transform requires, or the three-dimensional integral that the Fourier transform approach requires for each Slater orbital (with two additional integrals required, as in eq. ( 3) ).
Note that this new integral representation has a similar integrand to the integral representation introduced in our prior paper for products of M Slater orbitals, except that the new integral representation has finite limits of integration rather than the infinite interval of the previous work.

Problematic Approaches
The first step in creating the prior integral representation -and the new one -entails converting a product of Slater orbitals into denominators of some power (combined with other factors) using some initial integral representation, such as via the Stieltjes Transform [26], or [27], or [28], below In the prior paper we combined products of denominators into one, consolidating the coordinate variables into a common quadratic form, using [26] 1 (with p 1 = 1 and s = 2 for n = 1): One then performs the t integrals to obtain the n = 2 version of eq. ( 6).
To represent a product of M Slater orbitals using finite-interval integrals, one can in principle use Feynman parametrization [29] as extended by Schweber.[30] His third version of the extension, which may be derived by iterating [31,32] is n (10) so that, for instance, and this indeed bears fruit and is numerically stable up through M = 3.But the subsequent integration of the t s (as in the n = 2 version eq. ( 7)) produces functions whose arguments do not form discernable patterns as M increases, so we were not able to generalize this approach (using [28]) to large M.
In our prior work we used Fourier transforms to convert the product of M Slater orbitals into denominators instead of the integral set using [28] that we utilized in eq. ( 7).On a formal level, this approach worked fine with Schweber's third parametrization, yielding However, this was numerically stable only up through M = 3.After performing the ρ integral [23] to give the most compact form, the result was numerically stable only up through M = 2: This approach, then, should be reserved for analytical reduction of integrals rather than numerical integration.

Derivation of a Second, Numerically-Stable Integral Representation
Schweber's second parametrization, looks somewhat dubious for analytical uses since each succeeding integral has the prior parameter as its upper limit.It turns out, however, to give numerically stable results (we checked up through M = 6 when using the Fourier transform as the bridge rather than ( 7)), and one may perform a change of variables in each integral at the end of the derivation to give all integrals over [0,1].Since Fourier transforms include momentum variables in plane waves, we take the additional step of moving the combined momentum denominator into an exponential by using [33] Thus, for a product of M Slater orbitals, we have [26] e The quadratic form may be written as [12] where and Now suppose one could find an orthogonal transformation that reduced Q to diagonal form where, as shown by Chisholm, [34] the a ′ are positive.Then after a simple translation in space (with Jacobian = 1), the k integrals could be done, [22] Since this result is expressed in the form of an invariant determinant, actually finding the orthogonal transformation that reduces Q to diagonal form is unnecessary.What is left to find is just the exponential of −ρc ′ , which we integrate over ρ and the α i .
This orthogonal transformation also leaves Ω = detW (27) invariant and to find its value one need only expand Ω by minors: where Λ ij is Λ with the ith row and j th column deleted, and is diagonal in the present case.Therefore, c ′ (of eq. ( 24)) is given by so that We perform the ρ integral [23] to give the most compact, semi-final form for the desired integral representation: 5 Unifying the Upper Limits of the Integrals to Give a Third, Numerically-Stable Integral Representation The above forms are numerically stable but, for M > 2, their utility is somewhat hampered for analytical reduction since each succeeding integral has the prior parameter as its upper limit.One can cast each such integral into one over the interval [0, 1] by making a change of variables to in sequence from j = M − 1 down to j = 2 and by multiplying the set of derivatives of α j /α j−1 that defines each new variable σ j , The first three such are, where we explicitly put in the shifted coordinates R 2 j = x 2 1j (and = ´1 0 dα 1 ´1 0 dσ 2 ´1 0 dσ 3 This can easily be seen to generalize to (37) Inspection shows that for M = 2 we indeed obtain eq. ( 4).
One unusual feature of this integral representation is that, like that derived in the prior paper, the recursion relationships of Macdonald functions may be applied to lower (or raise) the indices.

Inclusion of plane waves and dipole interactions
Transition amplitudes sometimes contain plane waves, and these may be easily included in this integral representation directly in the ρ version, prior to completing the square, by utilizing an orthogonal transformation (like that for the k j ) that reduces the spatial-coordinate quadratic form to diagonal form.Again, one has invariant determinants for this orthogonal transformation so that it never needs to actually be explicitly found.As before, this is followed by a simple translation in {x 1 , x 2 , • • • , x N } space (with Jacobian = 1).
In the more compact version containing Macdonald functions, one can simply apply the translation in {x 1 , x 2 , • • • , x N } space to the plane wave(s) that multiply eqs.( 15), (31), and (37).
Photoionization transition amplitudes will generally contain dipole terms cos (θ) that may be transformed into plane waves via a transformation like cos θ , [35] giving an integrodifferential representation whose inclusion follows that for other sorts of plane waves.

Utilizing Meijer G-functions to reduce integrals
The utility of these new integral transformations for large M may well hinge on finding integrals over variables that reside within square roots as the argument of a Macdonald function.One method for crafting such untabled integrals is to violate a two general rules of procedure in analytic reduction of integrals.The first rule is to use sequential integration whenever possible.For instance, if one adds a third unshifted Slater orbital to eq. ( 1) and integrates over both variables, one would reasonably start by transforming only those Slater orbitals that contain x 1 and integrate over that variable.Next, one integrates the resultant and the third Slater orbital over x 2 .The result is easily found to be [36] S η10η120η20 1 (0, 0; 0, 0, 0 There is utility, however, in simultaneously transforming the full product of Slater orbitals to generate unusual integrals whose values we know (as above), but whose reduction path may be fraught with difficulty.If one can find the path for a known integral, this may provide a path for unknown integrals.And it is clear that the integral representations of the present paper, like that in the prior paper, are unusual in that they have integration variables residing within square roots as the arguments of Macdonald functions.
The present integral representation appears on the surface to be less likely to allow for such a reduction because there are many fewer tabled integrals over the interval [0, 1] than there are over the [0, ∞] interval in the integral representation of the prior work.We will see in applying this strategy to the above integral that this concern is not at all the case when one does both coordinate integrals first.
We apply the integral representation eq. ( 37) to all three Slater orbitals simultaneously.After completing the square and changing variables, the integral over x 1 + z 2 3/2 may be done using [24] and the consequent x 2 2 K 0 (ax 2 ) integral may also be done via [37] , and the second to last integral is given by [38] : following result to be integrated over: [40] 1 for which there is but one tabled integral [41] that has roughly the right form (with A modification was required since inserting α = 3 2 to remove the polynomial multiplying the G-function in the integrand leaves us with the wrong power of x.One may, however, take derivatives with respect to c of the integrand and resultant, with ν = 1/2 in combination with ν = 0, to show that where the reduction of the Meijer G-function in the third line is from [42] and the last step holds for a number of cases akin to the present one in which The integral representations of the current paper use integrals over the interval [0, 1] rather than over [0, ∞], and the only tabled integral over a similar G-function on [0, 1] we found [43] ˆ1 has a markedly different integrand than the square of the argument of the Macdonald function to which the fourth equality in (40) reduces, so it is useless for the present problem.Schweber's third parametrization gives no better result.All of this provided motivation for the integral representation we derive in the next section.

An Ungainly but Useful Bridge
There is an obscure integral [48] that will serve as an integral representation for a product of denominators, that has the very useful property of relating integrals over [0, ∞] to integrals over [0, 1] and [1, ∞].One might hope to extend integrals like [41] to integrals over [0, 1] or [1, ∞] if the integral over [0, ∞] could be found.
Its ungainliness is revealed in the process of extending it from a pair of denominators to triplets by iteration and beyond: at each step, the number of terms doubles so a general-M version is difficult to imagine.
To derive an integral representation for a product of two or three Slater orbitals, one simply follows the procedure laid out in Section 4, above, for each of the two or four terms, but the determinants are different in this new case and are different from term to term.The final forms are with the equality also holding for integrals over [1, ∞].It also holds over [0, ∞] if one multiplies the right-hand side by 1 2 1 2 .In the case of M ≥ 3, one may even mix these three intervals among the integrals present.We (simultaneously) apply the above integral representation (52) to all three Slater orbitals in the first line of (39), whose second line gives 117.4952904891590 when we abitrarily set parameters to {η 1 → 0.3, η 12 → 0.5, η 13 → 0.9}.After completing the square and changing variables, using [24] one may do the integral over (3 (M − 1) + M − 1) integral dimensions that the Fourier transform introduces for a product of M Slater orbitals.Direct integration of products of Slater orbitals containing angular functions centered on different points, usually bears fruit in only the simplest problems.The fourth conventional approach to these problems is to represent M Slater orbitals as M addition theorems, that is M infinite sums over Spherical Harmonics containing the angular dependencies.Orthogonality allows one to remove a few of these infinite sums in the process of integrating some of the original integrals.For large M, this approach rapidly bogs down.Each of the four extant approaches to such problems run into difficulties at some point as M increases.These three new integral representations (over the interval [0, 1]) for M Slater orbitals likewise have some positives and some negatives.We have found that only two of these three integral representations over the interval [0, 1] for M Slater orbitals is numerically stable for M > 3, though the other might be better for analytical work in some cases.The prior version having integrals running over the interval [0, ∞] was also numerically stable, but one does have the inconvenience of needing to test for a sufficiently large upper integration limit.
For the simplest problems, the three integral representations (over the interval [0, 1]) for M Slater orbitals of the present paper provide solutions in a much more rapid fashion than do the four extant approaches, and even surpass the integral representation over the interval[0, ∞] of the prior paper in allowing the moderately hard problem of the integral over three Slater orbitals -after all coordinate integrals have been done -to be reduced to analytical form via tabled integrals over the interval [0, 1].On the other hand, the integral representation over the interval [0, ∞] of the prior paper lacked any tabled result in the final step of this problem and had to rely on the computer algebra and calculus program Mathematica 7 to do this integral.This, however, belies the general paucity of tabled integrals over the interval [0, 1] relative to those over the interval [0, ∞].
The fact that the integration variables reside within a square root as the argument of a Macdonald function, shared by the prior work, will lead to difficulties in some complicated problems since only one such integral (transformed into a Meijer G-function) was known prior to the previous paper.Unlike that paper, the three integral representations for M Slater orbitals of the present work have the added difficulty of having no known integrals of this sort upon which to build.
It was for this reason that we introduced a fourth integral representation that is not easily generalizable to large M, but one hoped it would provide a bridge for finding the requisite integrals in the above problems.This final integral representation allowed us to derive the analytical result for an integral of a sum of two Meijer G functions f (x) G 2,0 0,2 ax 2 +b x+c x | 0, 0 over the interval [0, 1], via a bridge from the version of this integral representation that is over the interval [0, ∞].This is only half-way to the desired result, but is a promising step, and provides an integral that researchers in fields far afield from atomic theory may find useful.
0,2 ax 2 +b x+c x | 0, 0 over the interval [0, 1] for this research project to come to a sense of completion, I chanced upon an integral in Gröbner und Hofreiter [48] that would serve as an integral representation for a product of denominators.I found that I could use this as the basis for an integral representation for a product of several Slater orbitals that had the property of bridging from known integrals of Meijer G-functions (with such arguments) over the interval [0, ∞] to heretofore untabled pairs of such integrals over the the interval [0, 1] (and over [1, ∞]); a sort of mathematical Rosetta Stone.I have long had the practice of filling the 10 minutes prior to when my Astronomy class starts with videos of music featuring women instrumentalists, just as I make it my practice to bring video clips of experts in the field who happen to be women into the class content.The reader may or may not be aware that the historical predicament of women in STEM fields has significant parallels to the historical predicament of women in music, particularly when it comes to women instrumentalists.Indeed, women came to be auditioned into orchestras in significant numbers only after blind auditions were introduced in the 1970s.Neither did one see women instrumentalists playing with Miles Davis, say, or The Rolling Stones.Fortunately, both pop music and STEM fields are beginning to shift in this regard.It is my hope that as the younger generation comes to see women in both roles as "normal," they will help accelerate this shift.
So my students hear Lari Basilio shredding on electric guitar, [51] Sonah Jobarteh on the kora, [52] and Sophie Alloway on drums with Ida Hollis on electric bass, among many others.They likewise learn about the process of looking for life on Mars from Dr. Moogega Cooper, [54] and about the sound of Back Holes colliding from Dr. Janna Levin.[55] I share all of this detail so that it will be clear why the background soundtrack to my research into the material that comprises Sections 7 and 8 -and the idea that one sort of integral could act as a bridge or mathematical Rosetta Stone to craft others -was guitarist, composer, and singer Sister Rosetta Tharpe.[56] She was inducted by The Rock and Roll Hall of Fame as "the Godmother of Rock & Roll,"[57] though her influences on gospel, country, and R&B were also vast.
I am a jazz drummer who has been immersed in learning these other four musical styles over the past two decades, and, thus, Sister Rosetta Tharpe has been key to not only to deepening and generalizing my musical patterning, but also to the joy I experience in the process.On a weekly basis I am immersed in the creative expression of the musicians I jam with, and the rhythmical patterns they manifest in their music evoke a resonant rhythmical response in my drumming.Sometimes this response is delayed by months, because my skill-level needs to grow to accommodate it.And I sense, but cannot prove, that this response also manifests in my work as a theoretical physicist who relies heavily on pattern recognition for insights that culminate in my math-based results, such as the generalization to eq. ( 37) from the sequence from (34) to (36).
It is with all this in mind that I dedicate this paper, and in particuar the integral representations of Sections 7 and 8, to Sister Rosetta Tharpe.