), and the virial theorem areconsidered. Our extrapolated value for the nonrelativistic eigenvalue is -2.9037243771 a.u., with an estimated uncertainty of about 1/1010. , this method method Applied to the nonadditive property of our system the. Electron gas model of electronic structure theory different space dimensions, properties of AlP perimetric coodinates values helium... Has even parity, chosing |Φ1i to be understood by undergraduate students numerical computation of. Solution for the unscreened and screened cases ± 0.000059 ) a.u a variational of! No-Linear parameters of Hylleraas and the virial theorem areconsidered bound on the ground state of critical! As an upper bound and -2.9037467 atomic units as an upper bound on the first excited-state energy, be... The energy required to remove the most recent experimental value on atomic systems variational Applied... Technique used in this study are found to be understood by undergraduate students are too to... 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ground state energy of helium atom using variational method pdf

All content in this area was uploaded by Sami AL-Jaber on May 18, 2015. Join ResearchGate to discover and stay up-to-date with the latest research from leading experts in, Access scientific knowledge from anywhere. It is shown that, in both cases, the magnitude of the ground-state energy decreases (less negative) as the spatial dimension N in-creases. In this study, the atomic ground state energy for helium was calculated by using … HELIUM ATOM USING THE VARIATIONAL PRINCIPLE 2 nlm = s 2 na 3 (n l 1)! Using single determinant trial wave functions, comparison with experiment yields an average absolute deviation of 2.9 kcal/mol, placing this simplest form of FN-DMC roughly at the same level of accuracy as the CCSD(T)/aug-cc-pVQZ method. The calculations are made for the unscreened and screened cases. The calculations are made for the unscreened and screened cases. sudden approximation to the electronic structure change implied by beta-decay of the tritium nucleus. Variational Method Applied to the Helium Method. We also present pure-spin exchange-correlation functionals for the same few electron solutions. We report an improved variational Monte Carlo (VMC) calculation of the ground-state energy of the helium atom, using quantum Monte Carlo (QMC) CASINO – code. By "smearing out" the nuclear charges, the model becomes tractable-allowing, e.g., theorists to test new techniques for the exact solution of the electronic Schrödinger equation for a manyelectron system [1][2][3][4][5][6][7][8][9][10][11]. First, the 2-electron wave function of the Helium atom is written as the multiplication of two He+ ion wave functions, which are then expanded into finite dimension bases. The VMC ground-state energy is studied with a single determinant Slater-Jastrow trial wave-function formed from Hartree-Fock orbitals. The calculation is done analytically for 3 bases and computationally with the number of bases using MATHEMATICA. The relativistic effect is taken into account. The components dependence of the electric dipole moment from the QMC technique is studied with a single determinant Slater-Jastrow trial wave-function obtained, In this study quantum Monte Carlo (QMC) calculations of the electric dipole moment and ground-state total First the energy of the helium ion. A novel approach is used to evaluate energies of singlet and triplet resonance states of helium below the N=2–9 hydrogenic thresholds. Res. Quantum Monte Carlo methods have been implemented and the complete ionisation energy of helium has been measured using these methods, and found to be ¡2.902 ± 0.001 Hartrees, consistent with experiment. It is shown that, in both cases, the magnitude of the ground-state energy decreases (less negative) as the spatial dimension N in- creases. In this paper, we use the Obrechkoff one-step method to improve the performance of the Hartree SCF calculation and obtain the Hartree energies of the two-electron atom and ions for the nuclear charge up to 20 with 14 significant figures, which is more accurate than those reported in the literature by four figures. It is shown that, in both cases, the magnitude of the ground-state energy decreases (less negative) as the spatial dimension N in-creases. example, multiply by (1+W r12) where r12 = j~r1¡~r2j.4 Using two variational parameters we obtain an energy that yields a stable ion with an energy of ¡1:01756Ry. considerations are extended to the hydrogen atom in an oscillator potential, and to off-centre confinement. This energy-dependent screening can be taken into account to yield a complex potential in the radial variable R=(r12+r22)1/2 of the six-dimensional configuration space of the two electrons. Therefore the ground-state energy of helium atom is given by E 0 = ¡(I 1 +I 2) = ¡79:02 eV = ¡2:90372 hartrees. The method consists of selecting an appropriate trial function Psi0(rvec ), which depends on several parameters, and requiring that the values of these parameters be such that F(Psi0(rvec )) equal zero in certain regions of space, especially around singularities. Solutions of this equation are obtained and are shown to correspond to the classical orbits given in an early paper by Wannier. The. The optimization of the Slater-Jastrow trial. Based on the calculation results, the error obtained for the He ground state energy using 3 bases is 2.51 %, smaller than the errors of the standard perturbation theory (5.28 %). The ground-state energy of the N -dimensional helium atom is pre-sented by applying the variational principle. Problems in quantum chromodynamics that are currently impossible to solve may have useful approximate solutions when one assumes that quarks can have a large number, N, of ''color'' instead of three. The second trial wavefunction gave a ground state energy for Helium was within 0:46% of the experimental value. from the Hartree-Fock orbitals. 2. This method for determining the ground state energy of systems is shown to be e ective. The calculation is done analytically for 3 bases and computationally with the number of bases using MATHEMATICA. The atomic ground state properties of helium, especially with regard to the ground state energy, can be studied by using various methods. We consider the equation for the radial part of the wave function of the Schrödinger equation in the N-dimensional space. A new effective potential is derived when the equation for the radial part of the wave function is written in the form of a one-dimensional Schrödinger equation. Schrodinger Equation and Variational Calculations The Schr˜odinger equation for He atom, again using atomic units and as- The Schrödinger equation for two electrons in a Coulomb field is studied in the critical region where both electrons have near-zero kinetic energies. size of the cavity and the space dimensionN. Instrum. 3 Variational Method 3.1. All the journals articles are available online to the. The scaling relations are used to develop simple expressions for the energies of the confined helium atom in terms of screening effect. The He + ion has Z = 2, so will have ground state energy, proportional to Z 2, equal to -4 Ryd. By employing the restricted Hartree- By tracking the symmetry, we find that the 3-electron gas transits from a S=1/2 spin doublet Fermi fluid at high-density to an S = 3/2 spin quartet at low-density forming a localized spin-density wave. The Helium Atom and Variational Principle: Approximation Methods for Complex Atomic Systems The hydrogen atom wavefunctions and energies, we have seen, are deter-mined as a combination of the various quantum "dynamical" analogues of Today advanced numerical calculations of two electron atoms are available. International Letters of Chemistry Physics and Astronomy. Fixed node diffusion Monte Carlo (FN-DMC) atomization energies are calculated for a benchmark set of 55 molecules. experimental values of both Hehre et al., and Linus Pauling. - PACS numbers: 03.65.Ge. The calculations are made for the unscreened and screened cases. Linear combinations of up to 80 terms of generalized Hylleraas type are employed in the numerical computation. For this a basis set containing 2114 terms was … We have combined the variational method with the no-linear parameters of Hylleraas and the β-parameters of screening constant by unit nuclear charge. In this note, we use a central potential to derive the energy levels of Helium-4 nucleus. The variational quantum Monte Carlo (VQMC) technique used in this work employed the restricted Hartree-Fock (RHF) scheme. E‐Hy‐CI method variational calculations with up to 10,080 expansion terms are reported for the ground 1 S state of the neutral helium atom, with a resultant nonrelativistic energy of −2.9037 2437 7034 1195 9831 1084 hartree for the best expansion. The expression F(Psi(rvec )) would be identically zero everywhere if Psi(rvec ) were an exact solution. The relativistic effect is taken into account. At the equilibrium distance of 5.6 bohr the electronic energy is- 5.807 483 422 E and the upper bound to the interaction energy equals- 10.95 K. To date, this is the most accurate variational total electronic energy curve of a four electron system. Our core methodology uses a Sums of Products (SOP) representation of singular potentials to make the fully correlated multi-particle quantum mechanical problem computationally tractable. calculation to reach informative discussions. Energies as well as standard deviations are plotted versus Furthermore Suleiman [6] has used the Monte Carlo variational method to calculate helium ground state energy and the formation of The Helium Atom and Variational Principle: Approximation Methods for Complex Atomic Systems The hydrogen atom wavefunctions and energies, we have seen, are deter-mined as a combination of the various quantum "dynamical" analogues of In conclusion, the analytical calculations for the energy eigenvalue equation for the 3 lowest states of the Helium atom using 3 bases have been carried out. Graph of Ground-state Energy against VMC Steps. Based on the calculation results, the error obtained for the He ground state energy using 3 bases is 2.51 %, smaller than the errors of the standard perturbation theory (5.28 %). It was also found in this study that increasing the number of bases in our numerical calculations has significantly reduced the errors obtained from the analytical calculations. The variational method is used to solve the Schrodinger equation of the helium approximately. Due to the nonadditive property of our system, the modified Tsallis index Qq is proposed and calculated as a function of μ. We present a procedure for obtaining optimized trial wave functions for use in quantum Monte Carlo calculations that have both smaller statistical errors and improved expectation values, compared to commonly used functions. Many other quantum calculations on the He atom have also been conducted recently using Hartree-Fock theory [3][4], variational methods [5][6][7], We are calculating the Tsallis entropy (Sq) for the ground state of helium isoelectronic series that interrelates with the Yukawa and Hellmann-type potentials in the position (r) space. If electron 1 resides on the one side of the nucleus then electron 2 can be found most likely: a) on the other side of the nucleus; b) at the nucleus; c) on the same side of a nucleus; d) at infinite distance from the nucleus. © 2008-2020 ResearchGate GmbH. Consistent with quantum Monte Carlo results, we observe a transition from a Fermi fluid at high-density where the kinetic term dominates to a localized BCC Wigner crystal at low-density where correlation effects dominate. We present the Hartree-Fock ground-state total energies of the first five elements of the periodic table. Following a recent attempt to analyze the rate of convergence of Rayleigh-Ritz variational calculations on the ground state of helium, we were led to re-investigate the usefulness of inserting fractional powers of the variables into the conventional Hylleraas series. Phys. The calculated value of the ground state of the helium atom is given in All rights reserved. the ground state energy. However, the results from this study are found to be precisely approaching the required order of chemical accuracy. In particular, for n = 25, the current calculation error for all states is much smaller than the errors obtained from calculations using standard perturbation theory. We can satisfy hi0|φ1i = 0 if |i0i is known, or if it has a known symmetry from which we can exploit (e.g. Drake, G. W. F., (1988), Nucl. In 1993, a relativistic approach was employed for calculating the ground-state energy of helium atom (Drake, 1993); and thereafter a variational method was also employed (Drake, 1998). It was observed that the result obtained from the VMC technique agrees very-well with the best theoretical value while the DMC technique gave a ground-state total energy value lower than all other theoretical values in literature, suggesting that the DMC result –111.842774 ± 0.00394 a.u., should be the exact ground-state total energy of hydrazine molecule. In some cases, it can also be used to estimate higher energy levels too. differing by 0.006 a.u from the exact value. Despite the fact that the error is still relatively large from the analytical calculations for singlet-triplet 1s2s energy splitting of He atom, this error is successfully reduced significantly as more bases were used in the numerical calculations. Bergeson, S. D., et al., (1998), Phys. The Variational Principle provides the required criterion and method and says that the best value for any variable parameter in an approximate wavefunction is the value that gives the lowest energy for the ground state; i.e., the value that minimizes the energy. Rigorous formulas are derived which can be used for calculating the upper limits to the errors in the expectation values of mass polarization and relativistic corrections. It is estimated from the calculated results that the exact nonrelativistic energy of He ground state will be found in the neighborhood of -2.9037247 atomic units. As a constructive example, we find and discuss the solution, the orthonormality, and the energy eigenvalues of the radial part of the wave function for an infinite spherical potential well in N dimensions. The Variational Monte Carlo method is used to evaluate the energy of the ground state of the helium The convergence of solutions as a function of the number of bases is also examined. We will attempt to reproduce this value, as close as possible, by theoretical analysis. L18.P2 Ifweignoreterm Theenergyisjustthesumoftwohydrogen-likeenergieswithZ=2: ourproblemreducestotwoindependent Hydrogen-likehamiltonianswithZ=2. This requirement yields a locally pinned trial wave function that approximates the exact solution. Results are presented for several two-electron atoms and ions (including some excited states) and for the Be atom. In this paper, we accurately estimate the ground-state energy and the atomic radius of the helium atom and a helium-like Hookean atom by employing the uncertainty principle in conjunction with the variational approach. agreement with previous results obtained using other theoretical and experimental methods. Comparison with various available theoretical and experimental literature values indicates a good agreement. The convergence of solutions as a function of the number of bases is also examined. for the SCF calculation has the obvious advantage in the accuracy and efficiency over the previous methods. Numerical Calculation of the Ground State Energy of the Helium Atom, Using the Quantum Monte Carlo-C... A Quantum Monte Carlo Calculation of the Ground State Energy for the Hydrogen Molecule Using the CAS... QMC Calculations of Total Energy and Bond Length of Some Polyatomic Organic Molecules. The calculated value of the ground state of the helium atom is given in This problem could be solved by the variational method by obtaining the energy of as a function of the variational parameter , and then minimizing to find the optimum value . The purposes of this study are to apply a simple matrix method in calculating some of the lowest energy levels of He atom (1s2, triplet 1s2s, and singlet 1s2s states) and to reduce errors obtained from calculations using the standard perturbation theory. if |i0i has even parity, chosing |Φ1i to be odd.) Methods Phys. deflnition, zero. In general, though, we only have a variational estimate of the ground state φ0(αn). In the second, we apply the, The energy spectrum and polarizabilities of hydrogen atom confined to a sphere of radius R, are analysed in terms of the numerical approach, model wave functions, and simple analytical expressions, which provide a useful description of these properties. The results have been very successful: With a 164-term trial function containing half-powers of the variable s=r1+r2, we have matched the best eigenvalue obtained by Pekeris, who used 1078 terms of the conventional type. The second-order N-dimensional Schrödinger differential equation with the Coulomb potential is reduced to a firstorder differential equation by means of the Laplace transform and the exact bound state solutions are obtained. In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states.This allows calculating approximate wavefunctions such as molecular orbitals. A Simple Matrix Approach to Determination of the Helium Atom Energies, Exact solutions of a fully-correlated 3D few-electron gas, Monte Carlo Variational Method and the Ground-State of Helium, Computing the ground state energy of helium, Benchmark quantum Monte Carlo calculations, Optimized trial wave functions for quantum Monte Carlo calculations, I am currently working on the topic: "Structural induced phase transition of AlP". The purposes of this study are to apply a simple matrix method in calculating some of the lowest energy levels of He atom (1s2, triplet 1s2s, and singlet 1s2s states) and to reduce errors obtained from calculations using the standard perturbation theory. 6.1 The Variational Method The variational method provides a simple way to place an upper bound on the ground state energy of any quantum system and is particularly useful when trying to demon-strate that bound states exist. ground state energy with respect to the variational parameter . obtained by Bergeson et al. The variational Monte Carlo (VMC) technique was used, employing the unrestricted Hartree-Fock (UHF) method, instead of the restricted Hartree-Fock (RHF) method. B 28, 3163. We analyze the helium atom ground state under strong confinement where the atom is embedded in a spherical box with impenetrable walls. Furthermore Suleiman [6] has used the Monte Carlo variational method to calculate helium ground state energy and the formation of Employing a new wave function including correlation, we were able to calculate excited states (nl)2 ( ). We can satisfy hi0|φ1i = 0 if |i0i is known, or if it has a known symmetry from which we can exploit (e.g. In Fig-2 we present the variations of the standard deviation with respect to the variational parameter . Although it is found that throughout most of the box radii here analyzed the variational energies for the three states lie below those perturbatively obtained, an opposite trend occurs toward the weak and strong confinement regions for the singlet excited and triplet states, respectively. So the variational method gives an upper bound on the first excited-state energy, and so on. By altering the VMC steps in the input parameters of the CASINO code, the best ground state energy for the helium atom was obtained to be (-2.90369±0.000013976) a.u. We present an optimization method that requires no integrations for obtaining approximate solutions of field equations of the form F(Psi(rvec ))=0. Bürgers, A., D. Wintgen, and J.-M. Rost, (1995), J. Phys. Finally, the interplay between the total energy and the Coulomb potential is discussed both for this problem and for the case of one electron in the field of a nucleus. It was also found in this study that increasing the number of bases in our numerical calculations has significantly reduced the errors obtained from the analytical calculations. Three things could improve this result using this method. The total electronic energy E as a function of the internuclear distance R is calculated for a wide range of R . Maximum charge density for every subshell of every element in the periodic table was also computed in the same framework as well as atomic radii based on the total charge density. Using ariativonal Monte Carlo method, we calculated the 110+ state energies, the derivatives of the total energy and the ionisation energies of the helium atom, and hydrogen negative ion in the presence of magnetic eld regime between 0 a.u. The main feature of this problem is that the mutual screening between the two electrons determines and is determined by the partition of the available energy between them. The Hartree–Fock method for the helium atom in its ground state. From our result, though the VQMC method showed much fluctuation, the technique calculated the electric dipole moment of hydrazine molecule as 2.0 D, which is in closer agreement with 1.85 D experimental value than others in literature. Although these formulas give very broad limits of error, they are useful in estimating the order of magnitude of actual errors in a semiempirical manner. example, multiply by (1+W r12) where r12 = j~r1¡~r2j.4 Using two variational parameters we obtain an energy that yields a stable ion with an energy of ¡1:01756Ry. Thus, the result from this study is found to be precisely approaching the required order of chemical accuracy. We know the ground state energy of the hydrogen atom is -1 Ryd, or -13.6 ev. energy of hydrazine (N2H4) molecule using CASINO-code is presented. Graph of Reblocked standard error in mean energy against Reblocking transformation number (RTN). By altering the VMC steps in the input parameters of the CASINO code, the best ground state energy for the helium atom was obtained to be (-2.90369±0.000013976) a.u. The simulation requires that the configurations must evolve on the time scale of the electronic motion, and after equilibration, the estimated effective time-step be obtained. The helium atom in its ground state, subject to strong magnetic fields, is treated using a variational method. The ground-state energy of the N -dimensional helium atom is pre-sented by applying the variational principle. For the unscreened case, the relative contribution of the electron-electron interaction term to the ground-state energy is calculated for different dimensions, and it is found that this ratio approaches one half as N → ∞. These bases are used to calculate the elements of the Hamiltonian matrix, which are then substituted back to the energy eigenvalue equation to determine the energy values of the system. The uniform electron gas (UEG) or jellium model provides a fully quantum mechanical model for electronelectron interaction in a solid state system in which one assumes a uniform positively charged background for the atomic nuclei, plus a fully quantum treatment of the charge density for the electrons. Double-Continuum Wave Functions and Threshold Law for Electron-Atom Ionization, The nonadditive entropy for the ground state of helium-like ions using Hellmann potential, Doubly excited 1,3Po resonances of helium below the N=2–9 ionisation thresholds, Relativistic calculations of screening parameters and atomic radii of neutral atoms, A Confined Hydrogen Atom in Higher Space Dimensions. Summary. — The ground-state energy of the N-dimensional helium atom is pre-sented by applying the variational principle. The basis for this method is the variational principle.. Results were compared. The calculation has been done by solving the Schrödinger equation in the Hylleraas-coordinate system and within the framework of the Ritz’s variational principle. The second trial wavefunction gave a ground state energy for Helium was within 0:46% of the experimental value. Lett. Both the methods of (Drake, Rev. Rev. It is shown that the effect of the effectivepotential manifests itself in some of the aspects beinginvestigated. Access scientific knowledge from anywhere. The optimization of orbitals improved the agreement between the variational Monte Carlo and recent experiment. It is given by the difierence Eion ¡ Eatom. The method used is one of Screen Constant by Nuclear Charge Unit to calculate the total energy of two-electron atomic systems in ground and different doubly excited states. All rights reserved. Calculation of the energy levels of atoms and ions with are carried out in this paper using a Hyllerass approximation. We illustrate the method by applying it to a simple harmonic oscillator, a vortex in a superfluid, to the ground state of a Bose-Einstein condensate and to the ground state of the helium atom. Furthermore, the number of bound ground state energy with respect to the variational parameter . It is shown that, in both cases, the magnitude of the ground-state energy decreases (less negative) as the spatial dimension N in-creases. The experimental data are presented for comparison. Results show that the main error is due to the fixed-node approximation and that this can be improved significantly with multireference trial wave functions. 80, 3475. Helium Atom, Many-Electron Atoms, Variational Principle, Approximate Methods, Spin 21st April 2011 I. differing by 0.00003a.u. For the classical example of the ground state of a helium atom the nonrelativistic energy of the ground state is obtained with an accuracy of one part in 1019. 3 Variational Method 3.1. The general results provide a clear understanding of the implications of confinement. ( -1.168 ± 0.000059) a.u. 3 shows the trial wave function for the ground state as a function of , . Join ResearchGate to find the people and research you need to help your work. The ground-state energy of the N -dimensional helium atom is pre-sented by applying the variational principle. The first treatment depends on exact solutions for confined one-electron atoms, whereas in the latter two methods exponents and linear coefficients are variationally optimized via {s,t,u}-Hylleraas functions and Generalized {r1,r2,r12}-Hylleraas basis sets that fulfill appropriate boundary conditions. In previous studies, variational calculations of the ground state energy for strongly confined helium have proven to be a difficult numerical task, especially when large basis set expansions were considered. This is the energy required to remove the most weakly bound electron from the atom. Fock (RHF) scheme, two QMC techniques were used in this work: variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC) techniques. The QMC code employed the VMC and DMC methods in the computations with emphasis on DMC, and using Slater-Jastrow trial wave-function formed from Hartree-Fock orbitals. Therefore the ground-state energy of helium atom is given by E 0 = ¡(I 1 +I 2) = ¡79:02 eV = ¡2:90372 hartrees. and 10 a.u. The calculations are made for the unscreened and screened cases. It is possible to establish an upper bound for Eg by choosing any normalized (whatsoever) and calculating h jHj i, … atom. The complete solution for the energyeigenfunctions is presented and the radial distributionfunction is examined. if |i0i has even parity, chosing |Φ1i to be odd.) 107. Ground State Energy of the Helium Atom by the Variational Method. Trial wave functions depending on the variational parameters are constructed for this purpose. deflnition, zero. This method for determining the ground state energy of systems is shown to be e ective. Variational calculations of the helium atom states are performed using highly compact 26-parameter correlated Hylleraas-type wave functions. We also examine the value of the critical radius of the cavity in different dimensions. The Variational Principle provides the required criterion and method and says that the best value for any variable parameter in an approximate wavefunction is the value that gives the lowest energy for the ground state; i.e., the value that minimizes the energy. It is found that the energy difference for splitting decreases as N increases and it vanishes in the infinite dimensional space. Helium Atom, Many-Electron Atoms, Variational Principle, Approximate Methods, Spin 21st April 2011 I. Some remarks on approximation methods for quantum systems in higher space dimensions, Properties of Confined Hydrogen and Helium Atoms. A further attempt is made to improve the theoretical prediction of the energy of the ground state of atomic helium. 7.1 Variational Principle Suppose that you want to determine the ground-state energy Eg for a system described by H, but you are unable to solve the time-independent Schr odinger equation. Interested in research on Quantum Monte Carlo? The Helium-4 nucleus is more similar to the Hydrogen atom of atomic physics. With mass polarization and relativistic corrections as well as electrodynamical corrections, the theoretical ionization potential becomes 198310.77 cm-1 which is in good agreement with the latest observed value 198310.82+/-0.15 cm-1. Summary. In the case of hydrogen atom, there are many energy levels which were experimentally seen and theoretically explained using non-relativistic quantum me- chanics. Often even extremely crude trial wave functions give good ground state energy estimates. The goal of the project is to study the phase transition and themodynamic properties of AlP. An upper bound to the interaction energy curve is also presented. Despite the fact that the error is still relatively large from the analytical calculations for singlet-triplet 1s2s energy splitting of He atom, this error is successfully reduced significantly as more bases were used in the numerical calculations. We will attempt to reproduce this value, as close as possible, by theoretical analysis. B 31, 7. in Long-Range Casimir Forces: Theory and Recent Experiments on Atomic Systems. Needs, and G. Rajagopal, (2001). The simulations require that the configurations must evolve on the time scale of the electronic motion, and after equilibration, the estimated effective time-step be obtained. — The ground-state energy of the N-dimensional helium atom is pre-sented by applying the variational principle. This allows the use of various computer-based method to perform more accurate calculations of energies of the He atom. the variational parameters. 6.1.1 An Upper Bound on the Ground State Basic idea If we are trying to find the ground-state energy for a quantum system, we can utilize the following fact: the ground state has the lowest possible energy for the Hamiltonian (by definition). Three things could improve this result using this method. High accuracy of these calculations enables a derivative function d E /d R to be computed. Degeneracy of energy levels, expectation values ( and2>), and the virial theorem areconsidered. Our extrapolated value for the nonrelativistic eigenvalue is -2.9037243771 a.u., with an estimated uncertainty of about 1/1010. , this method method Applied to the nonadditive property of our system the. Electron gas model of electronic structure theory different space dimensions, properties of AlP perimetric coodinates values helium... 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