The J-Matrix Method Episode 11 pot

We obtain differential cross section in momentum space, provided the exit channel is described by the single HH Y ν0⍀.. 12 Function W ν0E forK = 0, 2 and 10 and l1= l2 = 0 The VP does no

d W ν05 (⍀) =Y ν0(⍀)2

d ⍀, dW5

ν0(⍀k)=Y ν0(⍀k)2

By analyzing the probability distribution, one can retrieve the most probable shape

of three-cluster shape or “triangle” of clusters A full analysis of a function of 5variables is non-trivial and one usually restricts oneself to some specific variable(s)

We integrate the probability distribution d W5

ν0(⍀) over the unit vectorsEq1,Eq2(resp

In coordinate space these can be interpreted as the squared distance between the

pair of clusters associated with coordinate q1, or, in momentum space, the relativeenergy of that pair of clusters We obtain

differential cross section in momentum space, provided the exit channel is described

by the single HH Y ν0(⍀)

In Fig 12 we display W ν0(E) for some HH’s involved in our calculations These

figures show that different HH’s account for different shapes of the three-cluster

systems For instance, the HH with K = 10 and l1 = l2= 0 prefers the two clusters

to move with very small or very large relative energy, or, in coordinate space, prefersthem to be close to each other, or far apart

4.2 Results

Again we use the VP as the N N interaction The Majorana exchange parameter m

was set to be 0.54 which is comparable to the one used in [53] The oscillator radius was set to b = 1.37 fm (as in [14, 19]) to optimize the ground state energy of the

alpha-particle

Fig 12 Function W ν0(E) for

K = 0, 2 and 10 and

l1= l2 = 0

The VP does not contain spin-orbital or tensor components so that total angular

momentum L and total spin S are good quantum numbers Moreover, due to the

specific features of the potential, the binary channel is uncoupled from the

three-cluster channel when the total spin S equals 1; this means that odd parity states

L π = 1−, 2−, will not contribute to the reactions.

To describe the continuum of the three-cluster configurations we considered all

HH’s with K ≤ Kmax = 10 In Table 11 we enumerate all contributing K -channels for L = 0 For each two- and three-cluster channel we used the same number

n = n ρ = Nint of basis functions to describe the internal part of the wave function

⌿L N int then also defines the matching point between the internal and asymptotic

part of the wave function We used N int as a variational parameter and varied it tween 20 and 75, which corresponds to a variation in coordinate space of the RGMmatching radius approximately between 14 and 25 fm This variation showed only

be-small changes in the S-matrix elements, of the order of one percent or less, and do not influence any of the physical conclusions We have then used N int = 25 for thefinal calculations as a compromise between convergence and computational effort

We also checked the impact of N int on the unitarity conditions of the S-matrix, for

instance the relation

S {μ},{μ}2+

ν0

S {μ},{ν0}2 = 1

Table 11 Number of Hyperspherical Harmonics for L= 0

We have established that from N int = 15 on this unitarity requirement is

satis-fied with a precision of one percent or better In our calculations, with N int = 25,unitarity was never a problem It should be noted that our results concerning theconvergence for the three-cluster system with a restricted basis of oscillator func-tions agree with those of Papp et al [58], where a different type of square-integrablefunctions was used for three-cluster Coulombic systems

In Fig 13 we show the total S-factor for the reaction3H3

H, 2n4

H e in the

energy range 0≤ E ≤ 200 keV One notices that the theoretical curve is very close

to the experimental data The total S-factor for the reaction3H e3

H e, 2p4H e

is displayed in Fig 14 It is also close to the available experimental data The

S-factor for both reactions is seen to be a monotonic function of energy, and does not

manifest any irregularities to be ascribed to a hidden resonance Thus no indicationsare found towards explaining the solar neutrino problem

The astrophysical S-factor at small energy is usually written as

S (E) = S0 + S

0E + S

We have fitted the calculated S-factor to this formula in the energy range 0 ≤ E ≤

200 keV For the reaction3H3

Fig 14 S-factor of the

exit channels for both6H e and6Be It is the Coulomb interaction that distinguishes

both systems, and accounts for the pronounced differences in the cross-sections and

in energy ranges between the calculated (0 ≤ E ≤ 200 keV) and experimental

(0≤ E ≤ 1000 keV) fits make it difficult to attribute any significant interpretation

to the discrepancy in the quadratic term

The HH’s method now allows to study some details of the dynamics of the tions considered In Figs 15 and 16 we show the different three-cluster

reac-K -channel contributions (W ν0 ) to the total S-factor of the reactions In Fig 15 these contributions (in % with respect to the total S-factor) are displayed for some fixed energy (1 keV), while Fig 16 shows the dependency of W ν0 (in absolutevalue) on the energy of the entrance channel One notices that three HH’s dom-inate the full result, namely the {K = 0; l1 = l2 = 0}, {K = 2; l1 = l2= 0} and

{K = 4; l1 = l2= 2}, and this is true in both reactions The contribution of these

states to the S-factor is more then 95% There also is a small difference between the

reactions3H3

H, 2n4H e and3H e(3H e, 2p)4H e, which is completely due to the

Coulomb interaction

Fig 15 Three-cluster channel contributions to the total S-factor for the reactions3H3

H , 2n4

H e

and 3H e3

H e , 2p4

H e in a full calculation with K max= 10

The Figs 15 and 16 yield an impression of the convergence of the results We

notice that the contribution of the HH’s with K > 6 is small compared to the

domi-nant ones This is corroborated in Fig 17 where we show the rate of convergence of

the S-factor in calculations with K max ranging from 0 up to 10 Our full K max = 10basis is seen to be sufficiently extensive to account for the proper rearrangement of

Fig 16 Three-cluster

channel contributions to the

total S-factor of the reactions

3H (3H , 2n)4H e in a full

calculation with K max= 10,

in the energy range

0≤ E ≤ 1000 keV

Fig 17 Convergence of the

S-factor of the reaction

3H (3H , 2n)4H e for K max

ranging from 0 to 10

two-cluster configurations into a three-cluster one, as the differences between resultsbecomes increasingly smaller

To emphasize the importance for a correct three-cluster exit-channel description,

we compare the present calculations to those in [50] , where only two-cluster figurations 4H e + 2n resp.4H e + 2p were used to model the exit channels In

con-both calculations we used the same interaction and value for the oscillator radius

In Fig 18 we compare both results for3H (3H, 2n)4H e An analogous picture is

obtained for the reaction3H e(3H e, 2p)4H e.

4.2.1 Cross Sections

Having calculated the S-matrix elements, we can now easily obtain the total and

differential cross sections In this section we will calculate and analyze one-folddifferential cross sections, which define the probability for a selected pair of clusters

to be detected with a fixed energy E12 To do so we shall consider a specific choice

Fig 18 Comparison of the

S-factor of the reaction

3H (3H , 2n)4H e in a

calculation with a

three-cluster exit-channel and

a pure two-cluster model

of Jacobi coordinates in which the first Jacobi vector q1is connected to the distance

between these clusters, and the modulus of vector k1is the square root of relative

energy E12 With this definition of variables, the cross section is

After integration over the unit vectors and substitution of sinθk, cosθk , d θkwith

d θk= 12

1

√

one can easily obtains d σ (E12)/d E12

In Fig 19 we display the partial differential cross sections of the reactions

3H3

H , 2n4H e and3H e3

H e , 2p4H e for the energy E = 10 keV in the trance channel The solid lines correspond to the case of two neutrons (protons)

en-Fig 19 Partial differential

cross sections of the reactions

3H (3H , 2n)4H e and

3H e(3H e , 2p)4H e

with relative energy E12, while the dashed lines represent the cross sections of the

α-particle and one of the neutrons (protons) with relative energy E12

We wish to emphasize the cross section in which two neutrons or two protonsare simultaneously detected One notices a pronounced peak in the cross section

around E12  0.5 MeV This peak is even more pronounced for the reaction

3H e3

H e, 2p4H e It means that at such energy two neutrons or two protons

could be detected simultaneously with large probability We believe that this peakcan explain the relative success of a two-cluster description for the exit channels at

that energy The pseudo-bound states of nn- or pp-subsystems used in this type of calculation then allows for a reasonable approximation of the astrophysical S-factor.

Special attention should be paid to the energy range 1-3 MeV in the4H e + n

and4H e + p subsystems This region includes 3/2−and 1/2−resonance states of

these subsystems with the Volkov potential In Fig 19 (dashed lines) we see that ityields a small contribution to the cross sections of the reactions3H3

H, 2n4H e

and3H e3

H e , 2p4H e This contradicts the conclusions of [53] and [54] where

the 1/2−state of the4H e + N subsystem played a dominant role We suspect this

dominance to be due to the interplay of two factors: the weak coupling betweenincoming and outgoing channels, and the spin-orbit interaction

In Fig 20 we compare our results for the total proton yield (reaction3H e3

H e,

2 p)4H e) to the experimental data from [66] The latter were obtained for incident energy E3

H e

= 0.19 MeV One notices a qualitative agreement between the

calculated and experimental data

The cross sections, displayed in Figs 19 and 20, were obtained with the maximal

number of HH’s (K ≤ 10) These figures should now be compared to the Fig 12,

Fig 20 Calculated and

Fig 21 Partial cross sections

for the reaction

3H e(3H e , 2p)4H e obtained

for individual K = 0, 2 and 4

components, compared to the

coupled calculation with

K max= 4 and the full

calculations with K max= 10

which displays partial differential cross sections for a single K -channel The cross

sections, displayed in Figs 19 and 20, differ considerably from those in Figs 12and comparable ones, even for those HH’s which dominate the wave functions ofthe exit channel An analysis of the cross section shows that the interference betweenthe most dominant HH’s strongly influences the cross-section behavior To supportthis statement we display the proton cross sections obtained with hypermomenta

K = 0, K = 2, K = 4 to those obtained with the full set of most important components K max ≤ 4 in Fig 21 One observes a huge bump around 10 MeV which

is entirely due the interference of the different HH components We also included

the full calculation (K max ≤ 10) to indicate the rate of convergence for this section

cross-5 Conclusion

In this chapter we have presented a three-cluster description of light nuclei on the

basis of the Modified J -Matrix method (MJM) Key steps in the MJM calculation

of phase shifts and cross sections have been analyzed, in particular the issue ofconvergence Results have been reported for6H e and6Be They compare favorably

to available experimental data We have also reported results for coupled two- andthree-cluster MJM calculations for the3H e3

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Other Related Methods: Chemical Physics

Application

Functional Theory with Auxiliary Basis Sets

Benny G Johnson and Dale A Holder

Abstract We present a generalized formulation of Kohn–Sham Density Functional

Theory (DFT) using auxiliary basis sets for fitting of the electron density that nificantly extends the range of applicability of this method by removing the currentcomputational bottleneck of the exchange-correlation integrals This generalizationopens the door to the development of a new fitted DFT method that is directly analo-

sig-gous to a J -matrix method, allowing the exchange-correlation energy and potential

of atomic and molecular systems to be calculated with an order of magnitude duction in computational cost and no loss in accuracy However, in contrast withprior approximate exchange-correlation methods, this computational advantage is

re-realized within a rigorous theoretical framework as with other J -matrix methods.

Generalized equations are presented for the self-consistent field energy, and ple applications are discussed In particular, it is shown that the stationary condi-tion of the energy with respect to the fitting coefficients can be removed withoutpenalty in complexity of the derivative theory, a characteristic drawback of mostfitted exchange-correlation treatments Results on accuracy and efficiency from animplementation of the new theory are presented and discussed

All quantum chemistry methods involve computing the following electronic totalenergy expression:

where the individual contributions are the one-electron, Coulomb, exchange andcorrelation energies, respectively The last term, arising from the correlation of themotions of the electrons to each other, is the most difficult to treat, and most often

is by far the most expensive part of a quantum chemistry calculation

In recent years, Density Functional Theory [1–3] (DFT) has emerged as an rate alternative first-principles quantum mechanical simulation approach in chem-istry, which is very cost-effective compared with conventional correlated methods.Once practiced in chemistry only by a small group of specialists, the last decade haswitnessed an explosion in growth in its usage, and DFT has become firmly estab-lished in mainstream chemistry research Perhaps the most compelling testament tothis fact is that the Nobel Prize in Chemistry in 1998 was awarded for work in DFT

accu-In several systematic validation studies DFT has exhibited good performance,and has often given results of quality comparable to or better than second-orderperturbation theory but at much lesser cost These encouraging results have providedincentive for the development of enhanced functionality within DFT programs Thecomputational attractiveness of DFT stems from its treatment of electronic exchangeand correlation (XC) at the self-consistent field (SCF) level via a functional of theone-electron charge density (and sometimes its derivatives), rather than requiring a

post-SCF calculation of correlation (e g perturbation theory, configuration

interac-tion), which is very expensive relative to the initial SCF procedure

The inclusion of correlation effects in an accurate fashion at the SCF level generallyimplies that the approximate density functional used in practice has a mathematicalform that is quite complicated Specifically, this has required computer implemen-tations of DFT for practical molecular calculations to resort to numerical quadra-ture to evaluate the exchange-correlation integrals involving the density functional.Sophisticated and accurate techniques have been developed for this purpose [4].The numerical calculation of the XC integrals must be approached mindfully,

as there are potential difficulties with grid-based methods (associated with tional and rotational invariance) that do not arise in methods where all the requisiteintegrals are evaluated analytically, for example, as in Hartree–Fock (HF) theory.However, with care these can be rigorously handled, as we have shown [5, 6] Giventhis, the single major drawback of the numerical integration scheme is its large com-putational cost

transla-One of the most important areas of research in modern quantum chemistry isthe continual search for ways to improve computational efficiency There is a com-pelling need to broaden the spectrum of applicability of these methods, in order tobring their powerful advantages to bear on as wide a range of chemical problems

as possible, maximizing their potential impact in research Before proceeding, it isimportant to note that the computational challenges facing quantum chemistry todayfall into two important categories: