Confined beryllium atom electronic structure and physicochemical properties

Confined beryllium atom ground and first excited states electronic structures are calculated by the direct variational method, taking into account the system asymmetric nature of the trial wave function, adding a cutoff function to ensure confinement boundary conditions. The trial wave function is built up from hydrogenic functions, which constitute an adequate basis for energies calculation. Physicochemical properties such as kinetic energy, pressure, and polarizability are also calculated from energy results previously obtained to different confined radii. Using different variational parameters in each hydrogenic function, the energy approximation obtained is improved. Electronic configuration changes as we move toward the strong confinement region (small cavity radii) in function of its atomic number using impenetrable walls, this region was obtained for Z = 4. This is a conclusion of this work. Another important result is that this method is computationally simpler and gives values inside the experimental precision. Aforementioned results are compared with other theoretical publications.


Introduction
Nowadays, it is of great interest to control and manipulate different systems properties, one way to achieve this is reducing space. When atom's electrons move is influenced by a potential barrier presence in at least one direction, it is said that the atom is confined. Confined quantum systems study began gaining importance around the 1930s, through a model proposal to study confined Hydrogen atom, located at a spherical box center with impenetrable walls; in order to determine its polarizability [1] variation as a pressure function.
It is well known that some system properties change when they are under spatial constraints effects, which may be either due to their size or to their particular environment; it is also possible that the system experiences restricted motion due to an external magnetic field presence. In many cases, system properties under such conditions may differ drastically with respect to those found in idealized or isolated systems. The reason why these changes occur can be found considering how most of the physical properties are implicitly related to the wave function and the energy, and this, in turn, is modified when the available space is restricted. So, to study this system type, it is generally necessary to find a solution for Schrödinger's equation using a Hamiltonian that includes space restrictions features.
The confinement model for atoms has also been used to study the electronic structure subjected to high pressures, as it has been for Helium atom [2][3][4][5][6][7] case; in effects of atoms and molecules trapped in nanostructures as fullerenes [8]; in multielectron systems such as atoms or molecules [9][10][11][12][13][14][15], as well as in quantum dots and quantum wires [16,17] study. There are other applications of this model in physics areas such as acoustics, solid state physics, nuclear physics, and biological studies in nanotechnology [18,19]. These systems physical properties characterization allows design and constructs devices such as ultra-small lasers, quantum light generators, specific wavelengths optical and electrical filters, among others; which are useful in modern electronics and optoelectronics. Energy studies, beryllium atom [20][21][22][23][24][25][26][27][28] lower excited states fine, and hyperfine structure play an important role in multielectron atoms excited states theory development, and better correlation effects understanding between electrons.
In this work, we present a theoretical characterization of a confined beryllium atom in a spherical box with impenetrable walls. The intent is to determine the effects due to confinement such physicochemical properties and electronic configuration. It was confirmed that electronic configuration of the ground state of confined beryllium atom, is different from that of the free beryllium atom depending on the confinement region. We proposed the direct variational method to do this calculation, which is computationally much simpler in comparison with FDT or Hartree-Fock, due to it requires fewer operations and it does not need a specialized software, pointing that we use a four-element basis; giving energy values inside the experimental precision. Using different effective atomic numbers for each hydrogen function, it is improved considerably the energy values precision due to electron screening effect consideration.
2. Variational study of the ground state and the first excited states of the beryllium atom into a spherical potential This section is dedicated to studying confined beryllium atom inside a box with spherical symmetry and impenetrable walls. Energies values will be obtained solving the time-independent Schrödinger equation using the direct variational method, based on the model proposed by Gorecki and Byers-Brown [29]. Hydrogenic functions will be used instead free-system functions, as well as introducing a cutoff function to ensure that the trial function is zero at the boundaries [5-13, 16-18, 30-32]. Considering a four-electron atomic system, the time-independent Schrödinger equation is given by the following eigenvalues equation where E, is the atom electron energy, Y is the wave function, and Ĥ is the Hamiltonian operator that depends on electrons coordinates. Hamiltonian operator for a four-electron atom in atomic units, using Born-Oppenheimer approximation and dismissing spin-orbit interaction, is given by: where V c , is the ansatz potential, limited by spherically symmetrical cavity and is defined by: V r r r r r r r r r r r r r r r , , , , 0, , , , , , , , , , 3 c 1 2 3 4 0 1 2 3 4 0 where r 0 , is confinement radius, and r i are electrons position vectors in the system, with i=1, K 4. A trial wave function will be used, in Slater's determinant form, using spin-orbital hydrogen functions for 1s and 2s orbitals where functions ξ(r i ) and χ(r i ), represent spin functions, j 1 (r) and j 2 (r) are hydrogenic functions for orbital 1s, while j 3 (r) and j 4 (r) are orbital 2s hydrogen functions. From previously performed variational approaches [14], we know that a better energy values approximation is obtained using different effective atomic numbers for different each orbital so that electron screening effect is taken into account. Therefore, we will use different variational parameters for each hydrogenic function constants, which are determined by the following condition: where E 0 is the lowest energy eigenvalue of Ĥ Using r r r r , , , After integrals calculation and plugged them into equation (11), it follows a numerical minimization process for each variational parameter (provided nuclear charge Z value and confinement radius r 0 ), namely: In order to improve confined atom energy approximate calculation, a slight modification was considered for the 2s functions, adding a different variational parameter to each of them, to give them more flexibility, being as follows: Once hydrogenic functions have been modified, 2s functions nodes are properly adjusted to reduce energy value. This change only affects integrals values, the energy functional form remains unchanged.
It is worth to remember that the variational method can be used to estimate excited states energy value, as long as it is ensured that the trial wave function is normalized and orthogonal to lowest states [33] wave function. Since 2p orbital has three projections 2p Z , 2p Y , 2p X ; in this work, the projections in z-direction and x-direction will be the one considered, the hydrogenic functions for that state are: In our case the trial wave function is orthogonal to the ground state function; this is due to hydrogenic functions angular part being orthogonal. The energy functional form remains unchanged, due to the change residing solely in the 2p orbital function, only modifying the integrals values. There is no angular dependence in the wave function ground state because orbitals only depend on the radial coordinate. Because of this, the Laplacian operator, which acts on the function, depends exclusively on the radial coordinate. Once 2p Z orbital with radial and angular dependence has been obtained, it is important to be very careful when calculating terms for kinetic energy so as not to make mistakes.
The so-obtained variational energy and the trial wave function, make it possible to calculate some confined beryllium atom properties. Average pressure exerted by system boundaries is given by the expression [16,18] is the sphere volume and E is the atom ground state total energy. To calculate kinetic energy, Ludeña [3] proposes the following equation, which relates kinetic energy K and pressure P with r 0 , as given by the virial theorem: An important physical quantity to calculate is polarizability, Kirkwood's [18] approximation was used: where a 0 , is Bohr radius, and α is polarizability.

Results and discussion
This section presents results associated with the variational method, as well as few confined beryllium atom physical properties, to describe pressure effect in the system electronic structure. Wolfram's mathematica software was used to optimize energy value for beryllium atom, using different confinement radii r 0 , provided the value for nuclear charge Z=4.

1s 2 2s 2 energy
Using four and six variational parameters, energy values and their respective variational parameters for beryllium atom's electronic configuration 1s 2 2s 2 are shown in tables 1 and 2, using the direct variational method and an antisymmetric wave function, where confinement radius r 0 is measured in Bohr and energy E H in Hartrees. The 1s orbital electrons experience higher nuclear charge than those in the 2s orbital. This is reflected in variational parameters values. As confinement radius decreases, system energy increases as expected, and the difference between values in variational parameters becomes smaller. A significant correction is observed in the energy using six variational parameters with respect to those obtained with four parameters. This is due to new parameters included in 2s hydrogenic functions, which allow for greater flexibility when energy value minimization is looked for.
Energy values comparison is shown in figure 1 using four parameters E , and those obtained by Ludeña [3] E SCF−HF , using a self-consistent field calculation.
Four and six variational parameters were used in this work. Compared to Ludeña's work [3], which used SCF approximation to the Hartree-Fock method, there was a 0.729% and 0.097% difference when r . 0  ¥ For remaining r 0 difference fluctuates between 1.122% and 2.854% using four parameters. Using six variational parameters, they fluctuate between 0.114% and 2.672%, reaching the largest difference in r 0 =1.25. Bohr for both cases. This difference has to do with the use of SCF approximation to the Hartree-Fock method, where the use of a sufficient number of basis functions is needed in order to calculate the analytical wave function precisely and, at the same time, to optimize orbital exponents, making calculations more complex and increasing computing time and effort. In contrast, in this work, we obtained sufficient energies to study a confined atom behavior with only six parameters. The use of such small base, composed of only four hydrogen-like functions, dramatically reduces calculation difficulty and execution time when minimizing energy values.
Compared to Rodriguez-Bautista's work [41], which used Roothaan's approach to solve the Hartree-Fock equations, there was a 0.097% difference when r 0 =10. They used a new basis set for Hartree-Fock calculations related to many-electron atoms confined by soft walls, and reported that orbital energies present one behavior totally different to that observed for confinements imposed by hard walls. Inner orbital energies do not necessarily go up when the confinement is applied, contrary to the increments observed when the atom is confined by walls of infinite potential. This is because for atoms with large polarizability, like beryllium and Potassium, external orbitals are delocalized when confinement is imposed. Consequently, internal oribatls behave as if they were in ionized atom.
Considering four ionization energies for beryllium atom, ground state energy is −14.669 3324 Hartrees, where a 1.318% difference was obtained for four variational parameters, and 0.690% for six when r 0 →∞. Nevertheless, before being able to compare a non-relativistic theoretically obtained value with experimental result, some additional effects have to be taken into account, such as nucleus movement with its finite mass (mass polarization), relativistic and radiative corrections and possibly nuclear charge distribution effect As expected, as confinement radius r 0 decreases, kinetic energy system increases due to the system pressure effect. This can be seen in table 3.
3.2. 1s 2 2p Z 2s and 1s 2 2p 2 energies First excited state experimental value for beryllium atom is unknown, therefore data obtained in this work will be compared to approximate results. The energy results obtained for different radius of confinement for the confined beryllium atom's electronic configuration 1s 2 2p Z 2s, as well as values obtained in different papers, are shown in table 4. It is evident from variational parameters that for the electron in 2 p orbital, the core is more shielded compared to other electrons. Its energy also rises when confinement radius decreases, same as ground state case. It is the largest, as well.  Table 3. Pressure and kinetic energy using (a) four variational parameters and (b) six variational parameters. Lower energy values obtained in this work have a 0.65% difference compared to those obtained by Hibbert [20], 0.64% compared to Weiss [21], and 0.92% compared to Chao Chen [35]. Hibbert and Weiss reported a set of large-scale configuration interaction (CI) calculations for the s snp n P 1 2 2 , 3 2 3 = ( ) states, which can give an accurate approximation for each state, but it may tend to obscure the global picture of the spectrum which is so transparent in the other approach. On the other hand, energies and wave functions for the beryllium atom are calculated with the full-core plus correlation wave functions by Chao Chen [35], obtaining a better approximation because of the use of many relevant angular and spin couplings which greatly contribute to the final energy values. Besides, Hibbert and Weiss did not include any intra-shell correlation in the s 1 shell, because their calculations were those of transitions in outer subshells. The purposes of these works were to obtain the energy values in a precise way though in our case we tried to find acceptable energy values to calculate atomic properties which were energy-dependent, plus, we consider the case of the non-free confined atom as Chao Chen [35], Hibbert [20] Weiss [21] did. All of which adds an additional potential due to confinement, which in turn influences on the difference among the values with respect to those ones already mentioned.
These methods are more expensive in terms of computation compared to the direct variational method because the CI basis sets expansion grows factorially and hundreds (sometimes thousands) of terms are needed in order to obtain the precision desired.
Montgomery [39], Dolmatov [44] and Saul Goldman [40] report that for a strong confinement regime the behavior of the orbitals is different from that in which the confinement is weak. For small confinement radii in the hydrogen atom, the energies of p 2 orbitals are lower than those of s 2 orbitals; energies different to the ones in the free atom. The crossing (intersection) of orbital energy for confined atoms was also reported by Garza et al [42], in particular for the Kr atom. It is well known that confinement overestimates the energies of the systems. Aquino et al [43] reported that a more physical way to simulate spherical compression would be accomplished by using soft, penetrable walls. Table 5 shows the energy results obtained for different radius of confinement for the beryllium atom with configuration s p 1 2 . 2 2 In figure 2, we show that the change in the beryllium atom's configuration confined to the ground state s s 1 2 2 2 takes place in the region r 0 2.3 0 < < Bohr radius, now being s p 1 2 2 2 the configuration for the ground state. Consequently, for r 2.3 0 < Bohr radius the energy values in tables 1 and 2 refer to the second excited state. The change in the electronic configuration of the beryllium atom, when r 0 decreases below 2.3, is one of the principal effects of spatial confinement [42], and can produce important changes in the physical properties such as electronegativity, softness, and hardness.
When an atom is confined, the energy of its ground state rises, as was showed above in figure 2. The same is true for the first and the second excited state, but the rise is much smaller. As a result, there is always a crossing point for cavities smaller than a critical size: the ground state of the atom lies higher in energy than the first confined states of the atom. Evidently, the ground state of the atom is no longer stable when it lies higher in Table 4. Comparison between energy values for confined beryllium atom's electronic configuration 1s 2 2p Z 2s and values obtained by Hibbert [20], Weiss [21], Chao Chen [35].

Beryllium atom polarizability
To calculate polarizability, Kirkwood's approximation [18] is used (equations (13) and (14)). Values obtained for free beryllium atom using different confinement radii r 0 and other reported results given by Komasa [36], Sahoo and Das [37] and Porsev and Derevianko [38], are shown in table 6. Polarizability is measured in units of a , 0 3 and as polarizability exact value is not reported for beryllium atom, we had to compare it to approximate data. For free atom case, we obtained a difference of 7.665%. Polarizability depends only on the box radius and varies monotonously with its radius as well. For a precise polarizability description, electronic correlation to a very  high level must be taken into account. Plus, a good outer region description is essential, thus electron density distribution becomes less important. Komasa et al [36], Sahoo-Das [37] and Porsev and Derevianko [38] calculated polarizability values with these aspects in mind. These requisites are met by very flexible wave functions that are explicitly correlated. In our case, we used a test function that is not the system's wave function.

Conclusions
The direct variational method, regardless of its complexity, turned out to be a simple and suitable approach from physicochemical and computational points of view. It is a method that allows saving computing time. Using six variational parameters we obtained better results than using four parameters, where the difference in energy values compared with Ludeña [3] results is 0.097% in case that the atom is not under any confinement potential r , 0  ¥ ( ) and between 0.114% and 2.672% in remaining cases. This difference is due to the Hartree-Fock, to accomplish major precision, needs enough basis functions, making more complex the calculation and increasing computing time. Whereas in our work, with only one basis formed by four elements we obtained a difference below 3.0%, which reduce computing time considerably.
In comparison with the beryllium atom experimental energy value, we obtained a 0.69% difference. And for this method simplicity used, this calculation can be implemented in personal computers, no requiring special conditions and computing time is less than an hour, which is a more efficient process than having a cluster.
There are only reported, by other authors, excited beryllium atom energy values in the ground state. We report here energy values to different confinement radii. The difference between reported energy values by cited author and ours for free atom, or non-confined, is 0.64% and 0.92%. No polarizability values have been reported for confined beryllium atom. In this work are reported values for this property with different confinement radii, where the difference between reported values for free atom and ours is 7.665%. In this work, we confirmed, as it was expected, that decreasing the confinement radii, energy, kinetic energy, pressure, and polarizability increased.
The energy functional expression presented in this work needed to calculate the energy values can be applied in further calculations regarding both free and confined beryllium and beryllium-like atoms with a base different from the hydrogenic one used in this paper.
Below Bohr radius refer to the ground state. Thus, it is possible to conclude that given a confinement Bohr radius of 2.3 for the beryllium atom, a change take place in the order the orbital energies in function of its atomic number.