1. Introduction
The theoretical studies of atomic systems in dense plasmas at different temperatures play a very important role in some physical situations and have gained considerable interest in recent years [
1,
2,
3,
4,
5,
6,
7,
8]. The dilute plasma environment is represented by the screened Coulomb potentials given by the Debye–Hückel model (DHM) or screened Coulomb potential (SCP) [
9], which provides a suitable treatment of nonideality in plasma via the screening effect under the low-density and high-temperature conditions. A closely related Hulthén potential is also used as a model potential for the dilute plasma environment in which the atoms are embedded. On the other hand, the dense quantum plasmas environment is represented by using the modified Debye–Hückel model (MDHM) [
10] or exponential cosine screened Coulomb potential (ECSCP). Due to its oscillatory nature, the MDHM potential represents a stronger screening effect than the DHM potential.
Considerable attention has been given to the screened Coulomb potentials and exponential cosine screened Coulomb potential in field theory, nuclear, and plasma physics [
11,
12,
13,
14,
15,
16,
17]. Accurate B-spline configuration interaction (BSCI) method was recently employed to study the spectral/structural data of the helium atom with exponential cosine screened Coulomb potentials [
18]. Roy [
19] discussed the critical parameter for the spherically confined H atom embedded within a diverse set of screened Coulomb potentials. Ghoshal and Ho [
20] investigated the two-electron system in the field of generalized screened potential within the framework of highly correlated and extensive wave functions in Ritz’s variational principle, where they were able to determine accurate ground state energies and wave functions of the two-electron system for different values of the screening parameter. Nasser, Zeama, and Abdel-Hady [
21] made a comparative study of the atomic Rényi and Shannon entropies with different wave function within the ECSCP for the 1s
2-state of the helium isoelectronic series in the Hylleraas-space with few variational parameters. Several other interesting studies on the few electron atoms embedded inside the different plasma potentials have been presented in the literature [
22,
23,
24]. Very recently a comprehensive compilation of accurate energy values and other structural parameters for the He-like atoms has been published [
25,
26] using the Hylleraas wave functions.
The purpose of this paper is twofold. Firstly, we report the energy values corresponding to the ground state and two low-lying excited electronic states of the He-like atoms embedded in three different plasma environments using the variational Monte Carlo (VMC) method [
27,
28,
29,
30,
31] and the Lagrange mesh method (LMM) [
32]. The second purpose of this work concerns with the application of the comparison theorem of quantum mechanics. According to the comparison theorem, if a set of spherical potentials V
1(r), V
2(r), V
3(r) satisfy the condition V
1(r) < V
2(r) < V
3(r) at all radial distances, then their eigenvalues obey E
1 < E
2 < E
3 for all nℓ-states. For nonrelativistic Hamiltonians bounded from below, this theorem follows directly from the variational characterization of the eigenspectrum [
33]. Generalized comparison theorems have been proposed [
34,
35,
36,
37,
38] that allow the two potential curves to cross over in a controlled fashion while maintaining a definite ordering of the respective eigenvalues. Refined comparison theorems applicable to the relativistic Dirac Hamiltonian, which is not bounded from below, have also been established [
39,
40,
41]. We refer to the works on the generalized comparison theorem [
42] and the refined comparison theorem [
43], which covers the current research trends in this area. In this work, we used the comparison theorem in order to rationalize the relative ordering of eigenspectra of the He atom under a set of different plasma screened Coulomb potentials for the ground and a few low-lying excited states. To the best of our knowledge, a comprehensive numerical test of the comparison theorem using the VMC and the LMM computations including the excited states, as reported in this work, has not been attempted earlier.
The outline of this paper follows. In
Section 2, we define the three different screened Coulomb model potentials that are introduced above. In
Section 3, an outline of the variational Monte Carlo (VMC) method employed in this work is presented. This is followed by the computational details and the choice of the trial wave functions for the low-lying excited states of He, described in
Section 4 and
Section 5, respectively. A brief description of the Lagrange mesh method employed in this work is presented in
Section 6. Our results are presented and discussed in
Section 7. Finally, the main conclusions of this work are listed in
Section 8.
2. Plasma Model Potentials
The collective effects of correlated many-particle interactions lead to screened Coulomb interactions in hot dense plasma conditions, which are commonly represented by the DHM or SCP and given by
where
represents the Debye screening parameter that determines the electronic interaction in the Debye plasma. It depends on the temperature and density of the plasma in the following form [
44]:
where
λD is called Debye screening length,
is the Boltzmann constant,
is the electron temperature,
e is the electronic charge,
Z is the atomic number, and
is the plasma–electron density. The Hulthén [
45] potential is given by
A useful form of Hulthén potential in which the screening parameter
μ in Equation (3) is simply multiplied by a factor of 2 can be defined as
It was shown that the study of effective screened potential in dense quantum plasmas can be represented by using MDHM [
10] or ECSCP, which is given by
Usually, in quantum plasmas,
is related to the quantum wave number of the electron, which is related to the electron plasma frequency. Furthermore, the definitions of
in the two model potentials are different. In the present paper, we are considering
as a parameter so that the physical difference of
between these model potentials [
14,
19,
46] is not discussed.
3. Variational Monte Carlo Method
Quantum Monte Carlo methods have already been used for quantum mechanical systems. There are several quantum Monte Carlo techniques such as VMC, diffusion Monte Carlo and Green’s function Monte Carlo methods. In this paper, we concentrate on the VMC method, which is used to approximate the eigenstate of the Hamiltonian
of a quantum mechanical system by some trial wave function
whose form is chosen from the analysis of the quantum mechanical system under study. Therefore, the expectation value of the Hamiltonian
is written as [
46]
where
is the local energy depending on the
coordinates
of the
electrons, and
is the normalized probability density. The variational energy can be calculated as the average value of
on a sample of
points
, sampled from the probability density
as follows:
In practice, the points
are sampled using the Metropolis–Hastings algorithm [
27,
28].
When evaluating the energy of the system it is important to calculate the standard deviation of this energy, given by [
47]
Since will be exact when an exact trial wave function is used, then the standard deviation of the local energy will be zero for this case. Thus, in the Monte Carlo method, the minimum of should coincide with a minimum in the standard deviation.
4. Theoretical Details
The nonrelativistic Hamiltonian in Hylleraas coordinates [
47] for the two electron systems, under effective SCP in dense plasmas is given, in atomic units, by
where
and
are the radius vectors of the twoelectrons relative to the nucleus, and
is their relative distance.
Moreover, the nonrelativistic Hamiltonian in the effective ECSCP is given by
The ground state of the helium atom is a spin singlet two-electron atom. Our calculations for this two-electron system depend on using an ultracompact accurate symmetric function, a nontrivial seven-parameter function, which is constructed by Turbiner et al. [
48] as follows:
with space wave function:
where
are
-dependent parameters and
is a permutation operator. This function leads for helium atom (
) to a certain improvement of the variational energy and the electron–nuclear cusp and at the same time, the electron–electron cusp. The function
represents the antisymmetric spin wave function with
spin up and
spin down as follows:
This function allows us to obtain the same relative accuracy in both cusp parameters and electronic correlation energy. The function appears as a uniform, locally accurate approximation of the exact ground state eigenfunction. It provides the same relative accuracies in energies and several expectation values together with both cusp parameters.
5. Trial Wave Functions for the Low-Lying Excited States of the Helium Atom
The study of the low-lying excited states of the helium atom has received considerable attention in theoretical investigations. Therefore, for the lowest two excited states, corresponding to the configurations 1s2s, we used the following trial wave functions:
- 1.
For the lowest ortho (space-antisymmetric) state 2
, corresponding to the configuration 1s2s, we considered the following simple trial wave function
- 2.
The state 2
is a para (space-symmetric) state corresponding to the configuration 1s2s and its trial wave function is, then, taken of the form
In these equations, are variational parameters and is the normalization constant. For spin functions, represents the singlet antisymmetric spin wave function with spin up and spin down as described in Equation (12).
The function
is the Jastrow correlation function given by [
49]
where
and
are variational parameters.
For the relationship of the electron–electron interaction, one obtains the cusp conditions
7. Results and Discussion
The numerical method used in our calculations, the VMC method, is based on a combination of the well-known variational method and the Monte Carlo technique of calculating the multidimensional integrals. By a suitable choice of the trial wave function, it is then possible to obtain minimum energy eigenvalues in agreement with the exact values for the ground as well as the excited states of the given atom. Accordingly, we investigated the effect of the plasma environment by using the SCP and the ECSCP models on the energy eigenvalues of the helium atom. The calculations are performed using a set of
Monte Carlo integration points to assess the accuracy with standard deviation of the order
. All our results are obtained in atomic units, i.e.,
. For the value of the ground state energy of the He atom that corresponds to Debye screening length
with screening parameter
and expresses the case of pure Coulomb potential, we obtained the value −2.902662 a.u., which nearly coincides with the value −2.9027 reported in [
48].
Table 1 shows the ground state energies of the helium atom under effective SCP in dense quantum plasma with the He
+ threshold energies and ionization potential
of He. The results show good agreement with the most accurate previous results, where the
parameter equals 2, and it is equivalent to the atomic nuclear charge for screening parameter
(
Debye screening length). For
, the parameter
starts to decrease slightly around the value 2; at
and at
,
.
In
Table 2, we present the ground state energies of the helium atom under effective ECSCP in dense quantum plasma. The He
+ threshold energies and ionization potential
of He are also given.
In
Table 3, we present the results of our calculations of the ground state energies of the helium atom under Hulthén potential in dense quantum plasma with the He
+ threshold energies and ionization potential
of He. For the one electron atoms, the energy ordering of
, has been a well-known consequence of the comparison theorem [
34]. More recently, based on a detailed mathematical analysis, a similar ordering has been conjectured for the He-like atoms [
57]. For the ground state He, the proposed conjecture has been validated numerically [
25,
56]. The energy data presented in
Table 1,
Table 2 and
Table 3 is employed in
Figure 1, where we display the variation of
,
, and
as a function of
. The adherence to the energy ordering
is numerically validated for the He atom in its ground state.
Another interesting ordering of energy levels for the one electron atoms is given by
, where
denotes the energy of the Hulthén potential at the screening parameter of
, as defined in Equation (4). As a natural extension of the conjecture [
56], this energy ordering is now tested numerically for the He atom in the ground and a few excited states.
In
Table 4, we list the VMC estimates of
,
,
, and
for
corresponding to the ground state He. The LMM estimates are given below the VMC estimates in each case. The two sets of values are found to be in good agreement with each other. The LMC estimates are uniformly below the VMC results. The latter values can in principle be improved further following a more accurate choice of the trial wave function. In
Table 4, we present the results of our calculations of ground state energy for He corresponding to the Hulthen (
μ), SCP, Hulthen (2
μ), and ECSCP potentials, given by Equations (1), (2), (4) and (5), respectively, over a representative set of μ values. In each case, we include the estimates obtained from the LMM and the VMC calculations. A numerical validation of the energy ordering
for the ground state He atom is evident from the data in
Table 4. It is clear from the present calculations that the inclusion of
makes the bounds to
tighter than given by the ordering without
.
In
Table 5 and
Table 6, we present the results of our calculations of
,
,
, and
corresponding to the (1s2s) excited states of
1S and
3S. As observed in the case of the ground state He in
Table 4, the VMC and the LMM estimates are in good agreement with each other and the comparison theorembased ordering
is also obeyed in the excited states. To the best of our knowledge,
Table 4,
Table 5 and
Table 6 present for the first time numerical validation of the conjecture
for the ground and excited states of the He atom.
From the Hamiltonian form of the two-body interactions, the overall potential strength decreases when going from pure Coulomb potentials to SCP and to ECSCP. Physically, we expect for screened potentials that the energy levels increase as
increases. Furthermore, due to stronger screening effects, for a given
, the ECSCP values should lie above the corresponding SCP data.
Figure 1 represents this situation.