New approach to obtain the analytical expression of the energy functional in free or confined atoms

Results in Physics 13 (2019) 102261

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New approach to obtain the analytical expression of the energy functional in free or confined atoms

T

A.D. Sañu-Ginartea, , E.M. Guillén-Romeroa, L. Ferrer-Galindob, L.A. Ferrer-Morenoc, Ri. Betancourt-Rierac, R. Rierab ⁎

a

Universidad de Sonora, Departamento de Física, Apartado Postal 1626, 83000 Hermosillo, Sonora, Mexico Universidad de Sonora, Departamento de Investigación en Física, Apartado Postal 5-088, 83190 Hermosillo, Sonora, Mexico c Instituto Tecnológico de Hermosillo, Avenida Tecnológico S/N, Colonia Sahuaro, C.P. 83170 Hermosillo, Sonora, Mexico b

ABSTRACT

A new approach to analytically generate the electronic energy functional expression for free or confined multi-electronic atoms via the Direct Variational Method (DVM) is presented. The electronic energy functional expression which is independent of both the basis set and confinement symmetry of the system is obtained through a new approach which is based on the orthonormality of spin functions, in which the DVM is applied. With this proposal the computation time is drastically reduced when getting the energy functional analytical expression for free or confined atoms either for its ground state or any excited state. The analytical expression to the electronic energy functional for Be atom generated in this work was compared with the expression for Be atom obtained in a previous work, where the DVM was used in its standard form. Due to lack of the electronic energy functional expression for the Li and B atoms, we were forced to calculate the energy values for these systems, to verify that the functional expressions generated in this work are correct and checked that its accuracy is within the results reported by other authors. In addition, a comparison between the response times is made by using the DVM in its standard form and the new approach proposed in this work.

Introduction The development of technology has been favored by the scientific community’s interest in the study of many-body systems (molecules, atoms or solids), subjected to certain confinement (soft [1–4] or hard walls [5–6], or in free state [7,8]). The infinite wall confinement model has been bound to the comprehension of how the electronic energy of these systems behaves and the effects that the confinement produces to their physical–chemical properties [9–11] and electronic structure [12]. These studies have shown that the electronic structures configuration and their physical–chemical properties either at ground or excited states, suffer significant changes under certain confinement radii [13], in contrast to their free state. Generally, an atom confined to infinite walls always increases all its orbital energies when the confinement radio reduces. Nevertheless, such observation is not always true for atoms confined to penetrable walls. The explanation to this result [1] is that, for atoms showing great polarizability, such as beryllium and potassium, the external orbitals delocalize when the confinement is established, causing the inner orbital energies not necessarily to increase. Efforts made by many researchers have contributed to a great deal of information to understand the physical phenomena, which take place

⁎

at the atomic level, giving rise to the possibility to develop new devices with novel properties and optimal performances. The most common models used for the study of these systems include variational methods [9,11–14], and Density Functional Theory (DFT) [15–17]. To solve the time-independent Schrödinger’s equation through DVM, we have:

E ( k) =

(r1, r2, …, rN , (r1, r2, …, rN ,

k) k)

H

(r1, r2, …, rN , (r1, r2, …, rN ,

k) k)

E0

(1)

where: (r1, r2, …, rN , k ) and (r1, r2, …, rN , k ) are the wave function and its conjugate. k is the variational parameters set, with k = 1, 2, … ri the position vectors of the electrons in the system (distance between the electron i and the nucleus). (r1, r2, …, rN , k ) H (r1, r2, …, rN , k ) is the Hamiltonian expectation value. (r1, r2, …, rN , k ) (r1, r2, …, rN , k ) is the norm. N is the electrons number. E ( k ) is the variational energy. E0 is the lowest system energy. H is the electronic Hamiltonian operator of N electrons in atomic units where it has been assumed that = m = e = 1 and the mass of the

Corresponding author. E-mail address: [email protected] (A.D. Sañu-Ginarte).

https://doi.org/10.1016/j.rinp.2019.102261 Received 31 October 2018; Received in revised form 12 March 2019; Accepted 30 March 2019 Available online 04 April 2019 2211-3797/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

Results in Physics 13 (2019) 102261

A.D. Sañu-Ginarte, et al.

nucleus is considered to be very large as compared with the electron’s mass (Born-Oppenheimer approximation) given by:

H =

1 2

N

N 2 i

Z

i=1

i=1

1 + ri

N

N

i=1 j=i+1

1 + Vc (r ) rij

method for the analytical finding of this functional, which is valid for free and confined atoms and independent from the base being used. The veracity of this method is proven with the analytical expression to the electronic energy functional for Be atom generated in this work with the expression for Be atom obtained by A. D. Sañu-Ginarte et al. [13], where the DVM was used in its standard form. In the case of Z = 3 and Z = 5, it was checked with the generation of the energy values under a confinement with spherical symmetry and impenetrable walls.

(2)

The first term represents the kinetic-energy for the electrons, the second one is the potential energy of attractions between the electrons and the nucleus, the third one is the potential energy of repulsions between the electrons and the last term is the confinement potential defined by:

Vc (ri, r0) =

0ifr1, r2, r3, , rN < r0 ifr1, r2, r3, , rN r0

Methodology In this section, the notation and the method to be followed to generate the analytical expression of the energy functional are presented, by means of our new approach.

(3)

where Z is the atomic number, rij is the interelectronic distance and r0 is the confinement radius, with i = 1, 2, , N and j = i + 1, , N , (N = Z for non-ionized atoms). A Slater-like determinant is used as a test wavefunction, which satisfies the anti-symmetrical condition of the wavefunction for fermions. All the elements within the same column in such a determinant refer to the same spin-orbital, while the other elements in the same row refer to the same electron. In the case of an infinite wall confinement it is necessary to assure that the test function cancels in the border. For this, it is necessary to introduce a function, called cutting function, in its linear [9], exponential [18], or potential [19] form.

(r1, r2, …, rN ,

=

1 N!

Notation The overlap integral Snm is given by:

Snm =

Jnm =

n (r1,

Knm =

1 (r1,

k)

(r1)

2 (r1,

k)

(r1)

…

N (r1,

k)

(r1)

1 (r2,

k)

(r2 )

2 (r2,

k)

(r2)

…

N (r2,

k)

(r2 )

(rN )

2 (rN ,

N (rN ,

k)

(rN )

k)

k)

(rN ) …

k )| m (r ,

(5)

k)

The Coulomb Jnm , and the interchange Knm integrals and the ones that are neither Coulomb nor interchange integrals Mnmkl are given by:

k)

1 (rN ,

n (r ,

n (r1,

Mnmkl = (4)

k ) m (r2,

n (r1,

k )|

k ) m (r2,

1 | r12

k )|

k ) m (r2,

n (r1,

1 | r12

k )|

k ) m (r2,

m (r2,

1 | r12

k (r1,

k)

k ) n (r1,

(6)

k)

k ) l (r2,

(7)

k)

(8)

The integral of the form Hnm which contain kinetic energy and interaction potential terms between electrons and the nucleus, are given by:

where functions (r ) and (r ) , represent spin functions, 1 (r , k ) , 2 (r , k ) and N (r , k ) are the function basis set. To make use of the DVM in the study of free or confined atoms after having calculated the determinant given by (Eq. (4)) and separated the spatial part from the spinorial part and taking advantage of the spin functions orthonormality, we proceed to the finding of an expression for the electronic Hamiltonian (Eq. (2)) expectation value. The expression that is obtained when calculating the Slater determinant, which grows factorially with respect to the electrons’ number, implies a difficulty in obtaining an electronic energy functional expression for atoms that contain a significant number of electrons, Z > 2 . For this, the use of parallelization techniques is needed, which consists in transforming a sequential program into a new equivalent semantic concurrent version in high performance distributed systems. With this work, it is possible to study free or confined atoms with considerably large Z values, due to the amount of the elements of our determinant non-zero is reduced. Many works have been focused on improving the energy values and not on optimizing the results obtaining process. In this work, the results on the investigation about atom electronic structures with Z = 3, 4 and 5 are presented. The results are based on the importance of optimizing the computing time, which grows factorially, when searching for the analytical expression of the electronic energy functional for free or confined multi-electronic atoms through DVM starting from a new approach to obtain the norm expression and the expression for the Hamiltonian expectation value. The form of the determinant proposed in this study, which constitutes our optimization, significantly reduces the computing time when searching out the analytical expression of the electronic energy functional. The i -th row in our determinant contains overlapping terms among the electron i and the other electrons j = 1, 2, , Z , cancelling those which contain different spin functions. The expression found after calculating the determinant being proposed is equivalent to the one obtained from , following the DVM approach. Our objective is to offer an optimal

Hnm =

n (r ,

k )|

2

2

|

m (r ,

k)

n (r ,

k )|

Z | r

m (r ,

k)

(9)

where n (r , k ), m (r , k ) , k (r , k ) and l (r , k ) represent the monoelectronic orbital functions, the range of the subscripts n , m , k and l is from 1 to N (N = Z for non-ionized atoms). Algorithm To analytically generate the expression of the energy functional of any atom through DVM, first, a test wavefunction is proposed as a Slater-like determinant. After having obtained the determinant expression, and having separated the spatial from the spinorial part, taking advantage of the spin functions orthonormality, we proceed to obtain the electronic Hamiltonian expectation value and the norm value. By means of this study, we save all this process, since the expression to calculate the determinant proposed is equivalent to the expression obtained to develop in the DVM, since all the interactions originating from the evaluation of the test wavefunction are already implicitly given in our determinant. Step 1 (Obtaining the norm expression) In this step, it is necessary to build up a determinant where each element represents an overlap integral between two spin-orbital functions (product of a monoelectronic spatial function by a monoelectronic spin), for example, the element nm , assuming the same spin orientation, is given by: nm

=

n (ri )| m (ri )

(ri)| (ri )

Snm

(ri )| (ri )

(10)

the range of both n and m is from 1 to N . In this way, the determinant which is not a Slater-like determinant for an N-electron atom is given by: 2

Results in Physics 13 (2019) 102261

A.D. Sañu-Ginarte, et al.

11

12

1N

21

22

2N

N1

N2

NN

J12 S33 S44 + J13 S22 S44 + J14 S22 S33 Relating the second term with the following terms:

J23 S11 S44 + J24 S11 S33

(11)

Relating the third term with the following terms:

This determinant for any given atom is formed following the electronic configuration under scrutiny, either for ground or excited states. When the spin functions are different, the position associated to the determinant has the value of zero, this is due to the spin functions orthonormality:

(ri )| (ri) =

(ri)| (ri ) = 1

(12)

(ri )| (ri) =

(ri )| (ri ) = 0

(13)

J34 S11 S22 From S13 S24 S31S42 , which is the second term of (eq. 15) , we get: Relating the first term with the successive terms:

M1234 S31 S42 + K13 S24 S42 + M1432 S24 S31 Relating the second term with the successive terms:

M2341 S13 S42 + K24 S13 S31

For example, for the Be atom in its ground state, the electronic configuration is (1s 2 2s 2 ) , two electrons in the 1s orbital and two in the 2s orbital. As two electrons occupying the 1s orbital have different spins, according to Pauli’s exclusion principle, the terms 12 and 21 become zero due to spin functions orthonormality, following this procedure, the terms 14 , 23 , 32 , 34 , 41 and 43 also become zero. After having taken advantage of the orthonormality from spin functions (eq. 12) and (eq. 13) , only two possibilities (0 or 1), the remaining ij terms are reduced to Sij , defining this way the Be atom determinant in its ground state as:

S11 0 S13 0 0 S22 0 S24 S31 0 S33 0 0 S42 0 S44

Relating the third term with the successive terms:

M3412 S13 S24 From S13 S22 S31S44 , which is the third term of (eq. 15) , we get: Relating the first term with the successive terms:

M1232 S31 S44

M2321 S13 S44

Relating the third term with the successive terms: From S11S24 S33 S42 , which is the last term of (eq. 15) , we obtain: Relating the first term with the successive terms:

(14)

M1214 S33 S42

S13 S22 S31 S44

M2343 S11 S42

S11 S24 S33 S42

M1412 S24 S33

K24 S11 S33

Relating the third term with the successive terms:

M3432 S11 S24 Step 3 (Obtaining the terms Hnm ) The Hij integrals generate when applying the following Hamiltonian operator terms:

Step 2 (Finding the elements Jnm , Knm, Mnmkl ) In the DVM approach, the following term is used:

i=1 j>i

J13 S24 S42

Relating the second term with the successive terms:

The final step consists in getting the energy functional from the norm expression. To achieve this, it is necessary to perform a series of transformations to obtain each of the terms produced when applying the Hamiltonian operator. These transformations are shown in steps 2 and 3.

N

J24 S13 S31

M3414 S13 S22

(15)

N

M1434 S22 S31

Relating the second term with the successive terms:

Then, the calculation for the determinant expression follows, coinciding with the one for the norm:

Norm = S11 S22 S33 S44 + S13 S24 S31 S42

K13 S22 S44

1 2

1 rij

N

N 2 i

Z

i=1

i=1

1 ri

To determine the Hnm integrals, each Snm element transforms into Hnm , being multiplied by the other terms Skl . For the Be atom: From S11S22 S33 S44 , which is the first term of (Eq. (15)), we get:

It represents the repulsion among electrons. From the electronic Hamiltonian (Eq. (2)) operator the test wavefunction integral of the type Jnm , Knm, Mnmkl can be obtained. These integrals act on different electron pairs. Because of this, we must get rid of the definition of the Snm integral and only take the subscripts into account. These elements in this method are determined taking each term from the norm calculated in the previous step and continuing in the following way: Each element of the same norm term is related with each of the following elements. 1. The following rule is applied, for any pair of elements.

H11 S22 S33 S44 + H22 S11 S33 S44 + H33 S11 S22 S44 + H44 S11 S22 S33 From S13 S24 S31S42 , which is the second term of (Eq. (15)), we get:

H13 S24 S31 S42 + H24 S13 S31 S42 + H31 S13 S24 S42 + H42 S13 S24 S31 From

S13 S22 S31S44 , which is the third term of (Eq. (15)), we get:

H13 S22 S31 S44 From

H31 S13 S22 S44

H44 S13 S22 S31

S11S24 S33 S42 , which is the last term of (Eq. (15)), we get:

H11 S24 S33 S42

if n = m and k = l, Jnk Snm Skl = if n = l and m = k, Knm else Mnkml

H22 S13 S31 S44 H24 S11 S33 S42

H33 S11 S24 S42

H42 S11 S24 S33

Step 4 (Functional generation) In this step, we must sum the expression obtained in Step2 and 3, and then divide by the norm considering:

2. The remaining terms Sij multiply the element Jnk , Knm or Mnkml , generated in the previous step. For our base example, the Be atom, Step two would yield: From the term S11S22 S33 S44 , which is the first term of (Eq. (15)), we obtain: Relating the first term with the successive terms:

Snm = Smn Hnm = Hmn Jnm = Jmn 3

Results in Physics 13 (2019) 102261

A.D. Sañu-Ginarte, et al.

Knm = Kmn

Table 1 Direct variational calculation for confined Lithium atom obtained in this work (ground and first excited state).

Mnmkl = Mnlkm = Mkmnl = Mklnm = Mmnlk = Mmkln = Mlnmk = Mlkmn Giving the analytical expression of the electronic energy functional for the Be atom in its ground state as follows:

Norm = S11 S22 S33 S44 + S132S24 2

S132S22 S44

1 E ( k) = [J12 S33 S44 + J13 S22 S44 Norm K24 S11 S33 + H11 S22 S33 S44

10 9 8 7 6 5 4 3 2 1

J13 S242 + J14 S22 S33 + J23 S11 S44

H11 S24 2S33

2

H22 S132S44

+ 2H13 S13 S24 + H22 S11 S33 S44 H44 S13 2S22 + 4M1234 S13 S24

K13 S22 S44 + K24 S132 2H13 S13 S22 S44 + 2H24 S132S24

H33 S11 S24 2 + H44 S11 S22 S33

2H24 S11 S24 S33 + H33 S11 S22 S44 2M1434 S13 S22

(16)

S11 S24 2S33

J24 S132 + J34 S11 S22 + K13 S24 2

+ J24 S11 S33

r0 (Bohr )

2M1214 S24 S33

2M2343 S11 S24 (17)

2M1232 S13 S44]

Results and discussion To verify the algorithm, we calculated the energies of some systems (Li , Be , B ), because there are no references of the functional expressions for these atoms and checked that its accuracy is within the results reported by other authors. From previously performed variational approaches [13], we know that a better energy values approximation is obtained using different effective atomic numbers for different each orbital so that each electron screening effect is taken into account. Therefore, different variational parameters will be used for each hydrogenic function

1

r r0

(18)

1

r r0

(19)

1 (r ,

k)

= n1 e

2 (r ,

k)

= n2 e

2r

3 (r ,

k)

= n3 (2

1 r 3 4 r)e 2

1

r r0

(20)

4 (r ,

k)

= n4 (2

1 r 5 6r)e 2

1

r r0

(21)

5 (r ,

k)

= n5 re

(

where 1

r r0

1 r 2 7 cos (

) 1

r r0

Confined Li atom, configuration

S11 0 S31 S41

(22)

+ H22 S11 S33

3)(Hartree ) first

2,

7.41923 7.41658 7.41444 7.40985 7.39970 7.37679 7.32300 7.18599 6.74635 5.08419 8.51392

7.35040 7.34330 7.33846 7.32975 7.31380 7.28376 7.22446 7.09747 6.78374 5.73861 2.32373

0 S13 S14 S22 0 0 0 S33 S34 0 S43 S44

(26)

S132S22 S44

S14 2S22 S33

S11 S22 S34 2 (27)

+ 2S13 S14 S22 S34 and

1s 22p1 2 p

E ( k) =

1 [J12 S33 S44 Norm k14 S22 S33

H22 S132 + H33 S11 S22

2M1232 S13]

K13 S22 S44 H11 S22 S34 2

2H13 S13 S22 S44 + 2H14 S13 S22 S34

2H14 S14 S22 S33 + H22 S11 S33 S44 + 2H22 S13 S14 S34 H22 S132S44

H44 S13 2S22

2H13 S13 S22

2H34 S11 S22 S34 + H44 S11 S22 S33

+ 2M1134 S22 S34 + 2M1232 S14 S34

2M1232 S13 S44

2M1242 S14 S33 + 2M1242 S13 S34

2M1314 S22 S34 + 2M1334 S14 S22

2M1343 S14 S22 + 2M1344 S13 S22

2M1434 S13 S22 + 2M2324 S13 S14

2M2324 S11 S34]

(25)

H22 S11 S34 2

+ H33 S11 S22 S44

H33 S14 2S22 +2H34 S13 S14 S22

(24)

K13 S22 + H11 S22 S33

J24 S132 + J34 S11 S22

K34 S11 S22 + H11 S22 S33 S44

+ 2H13 S14 S22 S34

(23)

S132S22

J12 S34 2 + J13 S22 S44 + J14 S22 S33 + J23 S11 S44

J23 S142 + J24 S11 S33

H22 S14 2S33

1 [J12 S33 + J13 S22 +J23 S11 Norm

E ( 1,

excited state 1s22p1 2 p

Norm = S11 S22 S33 S44

S11 0 S13 0 S22 0 S31 0 S33

E ( k) =

state

The functional expression obtained in the previous sections is valid for calculating the energy of a Be atom, both for its ground state 1s 2 2s 2 , and its first excited state 1s 2 2s12p1, which is not the lowest energy state. These results were published by A. D. Sañu-Ginarte et al. [13]. According to Hund’s rule [23], it can be predicted that for the first excited state of the Be atom, the lowest energy atomic state is 3P. For this state, and following the aforementioned algorithm, the determinant and the functional are given by:

For the Li atom the energy functional expression coincides both for its ground state 1s 2 2s1 2S and its first excited state 1s 22p1 2 p , only the expressions for the integral where the electron promoted plays a role will be different. The determinant and the functional have the following forms

Norm = S11 S22 S33

3, 4 )(Hartree ) ground

Confined Be atom, configuration 1s 22s12p1 3 p

) is the cutoff function. 1s 2 2s1 2S

2,

The experimental energy value for the Li atom [20] is 7.4786 hartrees for its ground state and 7.4106 hartrees for its first excited state. The values obtained in this study differ by 0.79% and 0.81% respectively. Sarsa et al. [21] calculated the Li atom energies for different basis sets, using the parameterized optimized effective potential (POEP) method, on the other hand, Ludeña [9] used a SCF Hartree-Fock, both confined the atom between impenetrable spherical walls. Comparing our results with theirs, a difference of about 0.2% was obtained. Comparing our results to Antonio Sarsa et al. [22], who used a Variational Monte Carlo method with Dirichlet boundary, a difference of about 0.8% was obtained. Near r0 = 3 bohr, as shown on Table 1, the energies of the first excited state are lower than the one for the ground state, this is due to the fact that the confinement breaks the orbital symmetry, making the 2p orbitals less energetic than 1s .

which is the same expression obtained by A. D. Sañu-Ginarte et al. [13], where DVM in its standard form was applied to get energy values for the Be atom for its ground and first excited states.

1r

E ( 1,

1s22s1 2S

(28)

From Table 2, it can be observed that for all the confinement radii, 4

Results in Physics 13 (2019) 102261

A.D. Sañu-Ginarte, et al.

Algorithm execution time

Table 2 Energy values of the first excited state obtained in this work for confined Beryllium atom. r0 (Bohr )

EH ( 1, 2, 3, 14.49477 14.49182 14.49041 14.48764 14.48140 14.46545 14.42037 14.28306 13.81933 11.82487 4.84199

10 9 8 7 6 5 4 3 2 1

4,

The aim of the Direct Variational Method (DVM) is to calculate the energy of a system using a trial wavefunction. The first step on the DVM is to calculate the analytical expression for the energy functional, and then, to calculate or obtain the smallest energy values optimizing the variational parameters. The improvement in the execution time shown in this study is linked to the acquisition of the analytical expression for the energy functional which is unique for a given configuration regardless of the base set selected. This improvement does not affect the energy values. Following the methodology used by A. D. Sañu-Ginarte et al. [13], to find the analytical expression of the electronic energy functional, the algorithm's response time for Li , Be and B was 0.00167 , 0.001 and 0.11933 minutes respectively, as shown in Table 4. With this work's proposal, described in the previous sections, the algorithm's response time got be drastically reduced, for example, for the B atom, it was possible to reduce the algorithm's delay time by 99.74 percent. In addition, it is valid to point out that with this method, the computational complexity curve remains low for Z < 6. The goal of this work is to improve the execution time to analytically determine the electronic energy functional expression for free or confined multi-electronic atoms, and not to improve the energy values. If it were the case, a way to improve these energy values would be to consider the spin–orbit interaction and electron-core interaction. Another important improvement would be to use a different set of basic functions with more variational parameters to give them more flexibility, when the process of numerical minimization begins. The energy values generated in this work are below those obtained by Ludeña [9], Sako and Diercksen [20], Sarsa et al. [21,22], because we ignore the electron-core interaction and do not contemplate the spin–orbit interaction. On the other hand, we only used basis function sets with two variational parameters. In all the systems studied in this work, the stronger the confinement, the greater the electronic energies. When the radii reduce, the confinement breaks the symmetry of the system, causing the atom prefers to ionize. The execution times to determine these functional expressions for the Li , Be or B in their ground or excited states, did not surpass 0.031 seconds, using a personal computer and without use parallelization techniques.

2 1 13 5)(Hartree ) first excited state 1s 2s 2p p

the energies of the [3]P state are lower than for the first excited state studied in A. D. Sañu-Ginarte et al. [13], reducing considerably the differences among the energies obtained by Hibbert [24], Weiss [25], Chao Chen [8] and A. D. Sañu-Ginarte et al. [13].

B atom confined, configuration 1s 22s 22p1 2 p and 1s 22s12p2 4 p The functionals obtained in this study for the B atom, both for its ground state and its first excited state following the algorithm described in Sections ‘Methodology’ are quite extensive. For this reason, we decided not to include them. The corresponding determinants for these states are given below. To demonstrate its validity, we compared the energy values shown in Table 3 with other theoretical studies.

S11 0 S31 0 S51

0 S22 0 S42 0

S13 0 S33 0 S53

0 S24 0 S44 0

S15 0 S35 0 S55

S11 0 S31 S41 S51

0 S22 0 0 0

S13 0 S33 S43 S53

S14 0 S34 S44 S54

S15 0 S35 S45 S55

(29)

(30)

Table 3 Direct variational calculation for confined Boron atom obtained in this work (ground and first excited state). r0 (Bohr )

EH ( 1,

10 9 8 7 6 5 4 3 2 1

24.50149 24.49788 24.49660 24.49444 24.49032 24.48088 24.45334 24.35253 23.91451 21.57930 1.44223

2, 3, 4, 5, 6,

7 )(Hartree ) ground

state 1s22s 22p1 2 p

EH ( 1,

2, 3, 4, 5, 6)(Hartree ) first

excited state 1s22s12p2 4 p

24.37529 24.37056 24.36892 24.36619 24.36107 24.34975 24.29505 24.18327 23.74374 21.56052 2.07631

Conclusions

According to Ludeña [9] for the B ground state, a difference between 0.11% and 0.16% was obtained as it can be seen in Table 3. For the first excited state, we did not find studies reporting the investigation of the confined atom.

A computational optimization to generate the energy functional analytical expression using the DVM approach was performed, to study confined or free multi-electronic atoms. Because the method evades the expansion of the Slater determinant, was possible to reduce the execution times in more than 90 percent in all cases. The functionals that have been applied to study the behavior of energies both for the ground 5

Results in Physics 13 (2019) 102261

A.D. Sañu-Ginarte, et al.

Table 4 Execution time comparison to generate the analytical expression of the electronic energy functional. Z

Execution time (minutes) In this work

Execution time (minutes) DVM standard [13]

Saved time (%)

3 4 5

1.7099E-5 4.425E-5 3.00533E-4

0.00167 0.01 0.11933

98.97 99.55 99.74

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and the first excited states of Li , Be and B atoms were obtained using an infinite confinement model with spherical symmetry. The precision achieved in this work, that is not our goal, for the energy values, are among the ones reported by other theoretical results, so we consider that our method can be used as a first approximation to determine a minimum mark for energy values in multi-electronic atoms. In addition, it is valid to point out that with this method, the computational complexity curve remains low for Z < 7. Acknowledgments This work was supported in part by CONACYT (Mexico) under contract 564944/297798. References [1] Rodriguez-Bautista M, Díaz-García C, Navarrete-López AM, Vargas R, Garza J. Roothaan’s approach to solve the Hartree-Fock equations for atoms confined by soft walls: basis set with correct asymptotic behavior. J Chem Phys 2015;143:034103. [2] Cruz SA, Díaz-García C, Olivares-Pilón H, Cabrera-Trujillo R. Many-electron atom confinement by a penetrable planar boundary. Radiat Eff Defects Solids 2016;171:123–34. [3] Montgomery HE, Sen KD. Dipole polarizabilities for a hydrogen atom confined in a penetrable sphere. Phys Lett Sect A Gen At Solid State Phys 2012;376:1992–6. [4] Aquino N, Flores-Riveros A, Rivas-Silva JF. Shannon and Fisher entropies for a hydrogen atom under soft spherical confinement. Phys Lett Sect A Gen At Solid State Phys 2013;377:2062–8. [5] Aquino N, Flores-Riveros A, Rivas-Silva JF. The compressed helium atom variationally treated via a correlated Hylleraas wave function. Phys Lett Sect A Gen At

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