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eISSN: 2576-4543

Physics & Astronomy International Journal

Research Article Volume 4 Issue 3

Calculation of singlet and triplet energy states of the two-dimensional (2D) H– ion and 2D He atom

NI Kashirina,1 Ya O Kashуrina,2 OA Koro,3 OS Roik2

1VE Lashkaryov Institute of Semiconductor Physics NAS of Ukraine, Ukraine
2Taras Shevchenko National University of Kyiv, Ukraine
3State University of Telecommunications Ukraine, Kyiv, Ukraine

Correspondence: Nataliya I Kashirina, VE Lashkaryov Institute of Semiconductor Physics NAS of Ukraine, 41 pr. Nauki, 03028, Kyiv, Ukraine

Received: May 14, 2020 | Published: May 29, 2020

Citation: Kashirina NI, Kash?rina YO, Korol OA, et al. Calculation of singlet and triplet energy states of the two-dimensional (2D) H– ion and 2D He atom. Phys Astron Int J. 2020;4(3):107-111. DOI: 10.15406/paij.2020.04.00207

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Abstract

Singlet and triplet energy states of the two-dimensional (2D) H and 2D He ions were calculated. А multi parameter system of Gaussian orbitals with exponentially correlated multipliers is used. An analog of the H ion is a two-electron shallow D center in covalent semiconductors. The energy of the lowest triplet term of 2D D center coincides with the bottom of the conduction band, which is a numerical illustration of Hill's theorem of the existence the only bound state for the hydrogen anion. The ground state energies and variational parameters for test wave functions are obtained. Useful limiting transition to the case of complete screening of the Coulomb repulsion Vee has been investigated. In this case, the Hamiltonian 2D H transforms into a two-dimensional hydrogen-like atom with two noninteracting electrons. The distribution of electrons by energy levels is carried out according to the Pauli principle. In a singlet state, the energy of such atom corresponds to the doubled ground state energy, in triplet one it is the sum of the energies of the ground and first excited states of the 2D hydrogen atom. The results are compared with the calculations performed by other authors. The energies obtained in the work with the use of Gaussian orbitals are the lowest in comparison with the results that have already been calculated by other authors for Slater type orbitals. This indicates a high accuracy of calculations with using Gaussian orbitals.

PACS numbers: 15.A- 31.15.xt 73.21.-b

Keywords: Gaussian orbitals, ground state energies, hydrogen anion, noninteracting electrons, Hamiltonian 2D Не

Abbreviations

1D, one-dimensional; 2D, two-dimensional; 3D, three-dimensional; MOC, metal oxide compounds; HTSC, high-temperature superconductivity

Introduction

Atoms and molecules with a reduced dimension can arise in large external magnetic fields. The magnetic traps were used by Görlitzet al1 in order to transfer sodium atoms to lower dimensional states. Transitions of sodium atoms in both two-dimensional (2D) and one-dimensional (1D) state were realized. Super strong magnetic fields can occur in the plasma of the Sun and stars. Therefore, in principle, one can observe the spectra of two-dimensional atoms and molecules in them. As is well known, on the Sun and Sun-like stars, the atoms of hydrogen and Helium lead to a small absorption of light. The main absorption provides a negative hydrogenion.2 Metal atoms make a small contribution to absorption, since their number is tens of thousands of times smaller than those of hydrogen atoms. Such a negative ion is formed when a second electron is attached to a hydrogen atom. The numerical research of anisotropic characteristics of a two-dimensional (2D) hydrogen atom induced by a magnetic field was carried out for Koval et al.3 Under terrestrial conditions, Н ions are unstable due to their extremely high chemical activity. A complete analog of the Hin semiconductor crystals is the Dcenter with a negative charge, i.e. a shallow hydrogen-like donor center that has captured an additional electron. The development of nanotechnologies has led to the emergence of new materials, such as two-dimensional monoatomic layers of various compositions. Graphene is a well-known example of a crystal with a two-dimensional hexagonal lattice in which one atom forms each vertex. There are other materials with a structure close to graphene.4,5 In such materials, it is possible to observe two-dimensional analogues of D centers in three-dimensional (3D) systems, as well as neutral two-electron states, similar to the He atoms.

Examples of two-dimensional systems are crystal structures that have translational symmetry in only two directions. Two-dimensional crystals can be located on the surface of bulk crystals, or on the surface of liquid solutions. Conductive layers in cuprate metal oxide compounds (MOC), in which high-temperature superconductivity (HTSC) was observed, can be considered as two-dimensional systems. Two-dimensional crystals have a band structure and can be both metals and semiconductors or dielectrics. Variational methods are used in the problems of quantum physics and chemistry devoted to the calculations of the energy spectrum of atomic and molecular systems. The energy functional of two-electron systems for 2D H and 2D He was obtained analytically using a Gaussian basis with exponentially correlated multipliers. Variational calculations were performed by random search methods and Hook-Jeeves method. The reasons why the random search method is preferred for optimization problems with a large number of parameters are described in review article.6 Hook-Jeeves method was used at the final stage to improve the accuracy of variational calculations.7

Basic formulas and relations

  1. Let's first look at a single-electron 2D atom. In atomic units (at. un.) (m=1, e=1, ħ=1), the Hamiltonian of a two dimensional atom with one electron has the form:

Н 1 = Δ 1 2 Z r 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGDq qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabg2da9iabgkHi TKqbaoaalaaakeaajugibiabfs5aeLqbaoaaBaaaleaajugWaiaaig daaSqabaaakeaajugibiaaikdaaaGaeyOeI0scfa4aaSaaaOqaaKqz GeGaamOwaaGcbaqcLbsacaWGYbqcfa4aaSbaaSqaaKqzadGaaGymaa Wcbeaaaaaaaa@4A3D@                               (1)

where, Z is the charge of the nucleus, the subscript 1 is introduced to denote a single-electron system, Δ1 – is the Laplace operator in a two-dimensional system:

Δ 1 = 2 x 1 2 + 2 y 1 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aenaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaeyOaIy7aaWbaaSqa beaajugWaiaaikdaaaaakeaacqGHciITcaWG4bWaa0baaSqaaiaaig daaeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWbaaSqa beaacaaIYaaaaaGcbaGaeyOaIyRaamyEamaaDaaaleaacaaIXaaaba GaaGOmaaaaaaGccaGGSaaaaa@4A20@

two-dimensional radius-vector is determined by the expressions:

r 1 =i x 1 +j y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahkhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaWHPbGaaGPaVlaadIhadaWgaaWc baGaaGymaaqabaGccqGHRaWkcaWHQbGaaGPaVlaadMhadaWgaaWcba GaaGymaaqabaaaaa@43B0@ , r 1 = x 1 2 + y 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaajugWaiaaigdaaSqabaGccqGH9aqpcaaMc8+aaOaaaeaacaWG 4bWaa0baaSqaaKqzadGaaGymaaWcbaqcLbmacaaIYaaaaOGaey4kaS IaaGPaVlaadMhadaqhaaWcbaqcLbmacaaIXaaaleaajugWaiaaikda aaaaleqaaaaa@484C@

here i, j – are the directing unit vectors in two-dimensional space.

The trial variational function is selected in the form:  

Ψ 1 = i=1 n C i exp( a 1i r 1 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaIaeuiQdK 1cdaWgaaqaaKqzadGaaGymaaWcbeaajaaicqGH9aqpkmaaqahajaai baGaam4qaOWaaSbaaSqaaiaadMgaaeqaaaqcbasaaiaadMgacqGH9a qpcaaIXaaabaGaamOBaaqcdaIaeyyeIuoajaaiciGGLbGaaiiEaiaa cchakmaabmaabaGaeyOeI0IaamyyamaaBaaaleaajugWaiaaigdaca WGPbaaleqaaOGaeyyXICTaamOCamaaDaaaleaajugWaiaaigdaaSqa aKqzadGaaGOmaaaaaOGaayjkaiaawMcaaaaa@5416@ ,                                               (2)

where ci , a1i are the variational parameters, a1i >0,

We introduce simplifying notation for the squared wave function (WF):

Ψ 1 2 ( r 1 )= i=1 n j=1 n c i с j exp( α ij r 1 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aa0 baaSqaaiaaigdaaeaacaaIYaaaaOWaaeWaaeaacaWHYbWaaSbaaSqa aiaaigdaaeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaabCaeaadaaeWb qaaiaadogadaWgaaWcbaGaamyAaaqabaGccaWGbrWaaSbaaSqaaiaa dQgaaeqaaOGaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsislcqaHXo qydaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamOCamaaDaaaleaacaaI XaaabaGaaGOmaaaaaOGaayjkaiaawMcaaaWcbaGaamOAaiabg2da9i aaigdaaeaacaWGUbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGym aaqaaiaad6gaa0GaeyyeIuoaaaa@591B@ ,

where α ij = a 1i + a 1j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadggadaWgaaWcbaGa aGymaiaadMgaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIXaGaam OAaaqabaaaaa@419A@ .

Normalization integral is:                  

N 1 = Ψ 1 2 ( r 1 )d σ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaaIXaaabeaakiabg2da9maapeaabaGaeuiQdK1aa0baaSqa aiaaigdaaeaacaaIYaaaaOGaaiikaiaahkhadaWgaaWcbaGaaGymaa qabaGccaGGPaGaamizaiaaho8adaWgaaWcbaGaaGymaaqabaaabeqa b0Gaey4kIipaaaa@4452@ ,

where d σ 1 =d x 1 d y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaaho 8adaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaWGKbGaamiEamaaBaaa leaacaaIXaaabeaakiaadsgacaWG5bWaaSbaaSqaaiaaigdaaeqaaa aa@4053@   is the element of the area in two-dimensional space.

Integration is performed in infinite limits in Cartesian coordinates:

{...} d σ 1 = {...} d x 1 d y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaaca GG7bGaaiOlaiaac6cacaGGUaGaaiyFaaWcbeqab0Gaey4kIipakiaa dsgacaWHdpWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Zaa8qCaeaaca aMe8+aa8qCaeaacaGG7bGaaiOlaiaac6cacaGGUaGaaiyFaaWcbaGa eyOeI0IaeyOhIukabaGaeyOhIukaniabgUIiYdaaleaacqGHsislcq GHEisPaeaacqGHEisPa0Gaey4kIipakiaaykW7caWGKbGaamiEamaa BaaaleaacaaIXaaabeaakiaadsgacaWG5bWaaSbaaSqaaiaaigdaae qaaaaa@5A16@ .

N 1 =π i,j=1 n c i c j α ij 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaaIXaaabeaakiabg2da9iabec8aWnaaqahabaGaam4yamaa BaaaleaacaWGPbaabeaakiaadogadaWgaaWcbaGaamOAaaqabaGccq aHXoqydaqhaaWcbaGaamyAaiaadQgaaeaacqGHsislcaaIXaaaaaqa aiaadMgacaGGSaGaamOAaiabg2da9iaaigdaaeaacaWGUbaaniabgg HiLdaaaa@4B60@ .

Kinetic energy is:

T 1 = 1 2 | Ψ 1 | 2 d σ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaWaa8qaaeaadaabdaqaaiabgEGirlabfI6aznaaBaaaleaacaaIXa aabeaaaOGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaakiaadsga cqaHdpWCdaWgaaWcbaGaaGymaaqabaaabeqab0Gaey4kIipaaaa@4874@ ,

T 1 =2π i,j=1 n c i c j a 1i a 1j α ij 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIXaaabeaakiabg2da9iaaikdacqaHapaCdaaeWbqaaiaa dogadaWgaaWcbaGaamyAaaqabaGccaWGJbWaaSbaaSqaaiaadQgaae qaaOWaaSaaaeaacaWGHbWaaSbaaSqaaiaaigdacaWGPbaabeaakiaa dggadaWgaaWcbaGaaGymaiaadQgaaeqaaaGcbaGaeqySde2aa0baaS qaaiaadMgacaWGQbaabaGaaGOmaaaaaaaabaGaamyAaiaacYcacaWG QbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa@50D1@ .

For potential energy we get:

V 1q =Z Ψ 1 2 ( r 1 ) r 1 d σ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaGaamyCaaqabaGccqGH9aqpcqGHsislcaWGAbWaa8qa aeaadaWcaaqaaiabfI6aznaaDaaaleaacaaIXaaabaGaaGOmaaaakm aabmaabaGaaCOCamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMca aaqaaiaadkhadaWgaaWcbaGaaGymaaqabaaaaaqabeqaniabgUIiYd GccaWGKbGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaaa@49B8@ ,

V 1q =Zπ π i,j=1 n c i c j α ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaGaamyCaaqabaGccqGH9aqpcqGHsislcaWGAbGaeqiW da3aaOaaaeaacqaHapaCaSqabaGcdaaeWbqaamaalaaabaGaam4yam aaBaaaleaacaWGPbaabeaakiaadogadaWgaaWcbaGaamOAaaqabaaa keaadaGcaaqaaiabeg7aHnaaBaaaleaacaWGPbGaamOAaaqabaaabe aaaaaabaGaamyAaiaacYcacaWGQbGaeyypa0JaaGymaaqaaiaad6ga a0GaeyyeIuoaaaa@4E83@ .

The energy of the ground state of a single-electron 2D atom has the form:

E 1 = min Ψ 1 ( T 1 + V 1q N 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIXaaabeaakiabg2da9maaxababaGaciyBaiaacMgacaGG UbaaleaacqqHOoqwdaWgaaadbaGaaGymaaqabaaaleqaaOWaaeWaae aadaWcaaqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWG wbWaaSbaaSqaaiaaigdacaWGXbaabeaaaOqaaiaad6eadaWgaaWcba GaaGymaaqabaaaaaGccaGLOaGaayzkaaaaaa@471A@ .

  1. Hamiltonian of two-electron 2D atoms has the form

Н 2 = Δ 1 2 Δ 2 2 Z r 1 Z r 2 + 1 r 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyhemaaBa aaleaacaaIYaaabeaakiabg2da9iabgkHiTmaalaaabaGaeuiLdq0a aSbaaSqaaiaaigdaaeqaaaGcbaGaaGOmaaaacqGHsisldaWcaaqaai abfs5aenaaBaaaleaacaaIYaaabeaaaOqaaiaaikdaaaGaeyOeI0Ya aSaaaeaacaWGAbaabaGaamOCamaaBaaaleaacaaIXaaabeaaaaGccq GHsisldaWcaaqaaiaadQfaaeaacaWGYbWaaSbaaSqaaiaaikdaaeqa aaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaG ymaiaaikdaaeqaaaaaaaa@4C86@ ,                          (3)

where r 12 =| r 1 r 2 | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcaaMc8+aaqWaaeaacaWH YbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaCOCamaaBaaaleaaca aIYaaabeaaaOGaay5bSlaawIa7aaaa@4314@

The variational functions of the singlet and triplet states of two-electron atoms are chosen in the form:

Ψ 12 ( r 1 , r 2 )= i=1 n 0 c i (1+ (1) S P 12 )exp( a 1i r 1 2 2 a 2i ( r 1 r 2 ) a 3i r 2 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aaS baaSqaaiaaigdacaaIYaaabeaakiaacIcacaWHYbWaaSbaaSqaaiaa igdaaeqaaOGaaiilaiaahkhadaWgaaWcbaGaaGOmaaqabaGccaGGPa Gaeyypa0ZaaabCaeaacaWGJbWaaSbaaSqaaiaadMgaaeqaaOGaaiik aiaaigdacqGHRaWkcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbe qaaiaadofaaaGccaWHqbWaaSbaaSqaaiaaigdacaaIYaaabeaakiaa cMcaciGGLbGaaiiEaiaacchadaqadaqaaiabgkHiTiaadggadaWgaa WcbaGaaGymaiaadMgaaeqaaOGaamOCamaaDaaaleaacaaIXaaabaGa aGOmaaaakiabgkHiTiaaikdacaWGHbWaaSbaaSqaaiaaikdacaWGPb aabeaakmaabmaabaGaaCOCamaaBaaaleaacaaIXaaabeaakiabgwSi xlaahkhadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacqGHsi slcaWGHbWaaSbaaSqaaiaaiodacaWGPbaabeaakiaadkhadaqhaaWc baGaaGOmaaqaaiaaikdaaaaakiaawIcacaGLPaaaaSqaaiaadMgacq GH9aqpcaaIXaaabaGaamOBamaaBaaameaacaaIWaaabeaaa0Gaeyye Iuoaaaa@6FAE@ ,                   (4)

where P12 is the operator of permutation of electron coordinates, S=0 for the singlet state, S=1 for the triplet state; r1, r2 are the radius vectors of the first and second electrons, respectively, ci, a1i, a2i, a3i are the variational parameters (a1i, a3i >0).

To simplify the calculations, the automatic WF symmetrization procedure is introduced. Always choose even n=2n0. For i > n0 we assume:

a 1i = a 3( i n 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIXaGaamyAaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaa iodadaqadaqaaiaadMgacqGHsislcaWGUbWaaSbaaWqaaiaaicdaae qaaaWccaGLOaGaayzkaaaabeaaaaa@40DA@ , a 2i = a 2( i n 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIYaGaamyAaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaa ikdadaqadaqaaiaadMgacqGHsislcaWGUbWaaSbaaWqaaiaaicdaae qaaaWccaGLOaGaayzkaaaabeaaaaa@40DA@ , a 3i = a 1( i n 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIZaGaamyAaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaa igdadaqadaqaaiaadMgacqGHsislcaWGUbWaaSbaaWqaaiaaicdaae qaaaWccaGLOaGaayzkaaaabeaaaaa@40DA@ .

For singlet states, we assume c i = c i n 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbaabeaakiabg2da9iaadogadaWgaaWcbaGaamyAaiab gkHiTiaad6gadaWgaaadbaGaaGimaaqabaaaleqaaaaa@3DDD@ , for triplet states c i = c i n 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbaabeaakiabg2da9iabgkHiTiaadogadaWgaaWcbaGa amyAaiabgkHiTiaad6gadaWgaaadbaGaaGimaaqabaaaleqaaaaa@3ECA@ .

Taking into account the automatic symmetrization of the WF, we omit the factor (1+(–1)sP12) and rewrite the two-electron WF in the form:

Ψ 12 ( r 1 , r 2 )= i=1 n c i exp( a 1i r 1 2 2 a 2i ( r 1 r 2 ) a 3i r 2 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aaS baaSqaaiaaigdacaaIYaaabeaakiaacIcacaWHYbWaaSbaaSqaaiaa igdaaeqaaOGaaiilaiaahkhadaWgaaWcbaGaaGOmaaqabaGccaGGPa Gaeyypa0ZaaabCaeaacaWGJbWaaSbaaSqaaiaadMgaaeqaaOGaciyz aiaacIhacaGGWbWaaeWaaeaacqGHsislcaWGHbWaaSbaaSqaaiaaig dacaWGPbaabeaakiaadkhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGc cqGHsislcaaIYaGaamyyamaaBaaaleaacaaIYaGaamyAaaqabaGcda qadaqaaiaahkhadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWHYbWa aSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaeyOeI0Iaamyyam aaBaaaleaacaaIZaGaamyAaaqabaGccaWGYbWaa0baaSqaaiaaikda aeaacaaIYaaaaaGccaGLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaa0GaeyyeIuoaaaa@653B@ .              (5)

We introduce simplifying notation for the squared wave function:

Ψ 12 2 ( r 1 , r 2 )= i=1 n j=1 n c i с j exp( α ij r 1 2 2 β ij ( r 1 r 2 ) γ ij r 2 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aa0 baaSqaaiaaigdacaaIYaaabaGaaGOmaaaakmaabmaabaGaaCOCamaa BaaaleaacaaIXaaabeaakiaacYcacaWHYbWaaSbaaSqaaiaaikdaae qaaaGccaGLOaGaayzkaaGaeyypa0ZaaabCaeaadaaeWbqaaiaadoga daWgaaWcbaGaamyAaaqabaGccaWGbrWaaSbaaSqaaiaadQgaaeqaaO GaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsislcqaHXoqydaWgaaWc baGaamyAaiaadQgaaeqaaOGaamOCamaaDaaaleaacaaIXaaabaGaaG OmaaaakiabgkHiTiaaikdacqaHYoGydaWgaaWcbaGaamyAaiaadQga aeqaaOWaaeWaaeaacaWHYbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXIC TaaCOCamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiabgkHi Tiabeo7aNnaaBaaaleaacaWGPbGaamOAaaqabaGccaWGYbWaa0baaS qaaiaaikdaaeaacaaIYaaaaaGccaGLOaGaayzkaaaaleaacaWGQbGa eyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaSqaaiaadMgacqGH9a qpcaaIXaaabaGaamOBaaqdcqGHris5aaaa@7152@

where α ij = a 1i + a 1j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadggadaWgaaWcbaGa aGymaiaadMgaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIXaGaam OAaaqabaaaaa@4112@ , β ij = a 2i + a 2j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadggadaWgaaWcbaGa aGOmaiaadMgaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIYaGaam OAaaqabaaaaa@4116@ , γ ij = a 3i + a 3j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadggadaWgaaWcbaGa aG4maiaadMgaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIZaGaam OAaaqabaaaaa@411E@ , α ij >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadMgacaWGQbaabeaakiabg6da+iaaicdaaaa@3B6B@ , γ ij >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaadMgacaWGQbaabeaakiabg6da+iaaicdaaaa@3B73@ .

Normalization integral:

N 2 = Ψ 2 ( r 1 , r 2 )d σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaaIYaaabeaakiabg2da9maapeaabaGaeuiQdK1aaWbaaSqa beaacaaIYaaaaOWaaeWaaeaacaWHYbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaahkhadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaa caWGKbGaaC4WdmaaBaaaleaacaaIXaGaaGOmaaqabaaabeqab0Gaey 4kIipaaaa@4721@

where d σ 12 =d r 1 d r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaaho 8adaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Jaamizaiaahkha daWgaaWcbaGaaGymaaqabaGccaWGKbGaaCOCamaaBaaaleaacaaIYa aabeaaaaa@4083@ .

N 2 = π 2 i,j=1 n c i c j ς ij 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaaIYaaabeaakiabg2da9iabec8aWnaaCaaaleqabaGaaGOm aaaakmaaqahabaGaam4yamaaBaaaleaacaWGPbaabeaakiaadogada WgaaWcbaGaamOAaaqabaGccqaHcpGvdaqhaaWcbaGaamyAaiaadQga aeaacqGHsislcaaIXaaaaaqaaiaadMgacaGGSaGaamOAaiabg2da9i aaigdaaeaacaWGUbaaniabggHiLdaaaa@4C5A@

where ς ij = α ij γ ij β ij 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOWdy1aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iabeg7aHnaaBaaaleaa caWGPbGaamOAaaqabaGccqaHZoWzdaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyOeI0IaeqOSdi2aa0baaSqaaiaadMgacaWGQbaabaGaaGOm aaaaaaa@4775@ , ς ij >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOWdy1aaS baaSqaaiaadMgacaWGQbaabeaakiabg6da+iaaicdaaaa@3B71@ .

Kinetic energy is:

T 2 = 1 2 ( | 1 Ψ 12 | 2 + | 2 Ψ 12 | 2 ) d σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIYaaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaWaa8qaaeaadaqadaqaamaaemaabaGaey4bIe9aaSbaaSqaaiaaig daaeqaaOGaeuiQdK1aaSbaaSqaaiaaigdacaaIYaaabeaaaOGaay5b SlaawIa7amaaCaaaleqabaGaaGOmaaaakiabgUcaRmaaemaabaGaey 4bIe9aaSbaaSqaaiaaikdaaeqaaOGaeuiQdK1aaSbaaSqaaiaaigda caaIYaaabeaaaOGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaaaO GaayjkaiaawMcaaaWcbeqab0Gaey4kIipakiaaykW7caWGKbGaaC4W dmaaBaaaleaacaaIXaGaaGOmaaqabaaaaa@57B6@ ,

T 2 =2 π 2 i,j=1 n c i c j ς ij 2 [ α ij ( a 2i a 2j + a 3i a 3j )+ γ ij ( a 1i a 1j + a 2i a 2j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIYaaabeaakiabg2da9iaaikdacqaHapaCdaahaaWcbeqa aiaaikdaaaGcdaaeWbqaaiaadogadaWgaaWcbaGaamyAaaqabaGcca WGJbWaaSbaaSqaaiaadQgaaeqaaOGaeqOWdy1aa0baaSqaaiaadMga caWGQbaabaGaeyOeI0IaaGOmaaaakmaadeaabaGaeqySde2aaSbaaS qaaiaadMgacaWGQbaabeaakmaabmaabaGaamyyamaaBaaaleaacaaI YaGaamyAaaqabaGccaWGHbWaaSbaaSqaaiaaikdacaWGQbaabeaaki abgUcaRiaadggadaWgaaWcbaGaaG4maiaadMgaaeqaaOGaamyyamaa BaaaleaacaaIZaGaamOAaaqabaaakiaawIcacaGLPaaacqGHRaWkcq aHZoWzdaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaeWaaeaacaWGHbWa aSbaaSqaaiaaigdacaWGPbaabeaakiaadggadaWgaaWcbaGaaGymai aadQgaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIYaGaamyAaaqa baGccaWGHbWaaSbaaSqaaiaaikdacaWGQbaabeaaaOGaayjkaiaawM caaiabgkHiTaGaay5waaaaleaacaWGPbGaaiilaiaadQgacqGH9aqp caaIXaaabaGaamOBaaqdcqGHris5aaaa@7272@

β ij ( a 1i a 2j + a 2i a 1j + a 3i a 2j + a 2i a 3j ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamGaaeaacq GHsislcqaHYoGydaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaeWaaeaa caWGHbWaaSbaaSqaaiaaigdacaWGPbaabeaakiaadggadaWgaaWcba GaaGOmaiaadQgaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIYaGa amyAaaqabaGccaWGHbWaaSbaaSqaaiaaigdacaWGQbaabeaakiabgU caRiaadggadaWgaaWcbaGaaG4maiaadMgaaeqaaOGaamyyamaaBaaa leaacaaIYaGaamOAaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaik dacaWGPbaabeaakiaadggadaWgaaWcbaGaaG4maiaadQgaaeqaaaGc caGLOaGaayzkaaaacaGLDbaaaaa@5681@ .

In order to calculate the potential energy, it is convenient to introduce the following Fourier components of the two-electron wave function:

W 12q = Ψ 12 2 exp(iq r 12 ) d σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaaIXaGaaGOmaiaadghaaeqaaOGaeyypa0Zaa8qaaeaacqqH OoqwdaqhaaWcbaGaaGymaiaaikdaaeaacaaIYaaaaOGaciyzaiaacI hacaGGWbGaaiikaiaadMgacaWHXbGaaCOCamaaBaaaleaacaaIXaGa aGOmaaqabaGccaGGPaaaleqabeqdcqGHRiI8aOGaamizaiaaho8ada WgaaWcbaGaaGymaiaaikdaaeqaaaaa@4DA1@ ,

W 1q = Ψ 12 2 exp(iq r 1 ) d σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaaIXaGaamyCaaqabaGccqGH9aqpdaWdbaqaaiabfI6aznaa DaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGcciGGLbGaaiiEaiaacc hacaGGOaGaamyAaiaahghacaWHYbWaaSbaaSqaaiaaigdaaeqaaOGa aiykaaWcbeqab0Gaey4kIipakiaadsgacaWHdpWaaSbaaSqaaiaaig dacaaIYaaabeaaaaa@4C29@ ,

W 2q = Ψ 12 2 exp(iq r 2 ) d σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaaIYaGaamyCaaqabaGccqGH9aqpdaWdbaqaaiabfI6aznaa DaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGcciGGLbGaaiiEaiaacc hacaGGOaGaamyAaiaahghacaWHYbWaaSbaaSqaaiaaikdaaeqaaOGa aiykaaWcbeqab0Gaey4kIipakiaadsgacaWHdpWaaSbaaSqaaiaaig dacaaIYaaabeaaaaa@4C2B@ ,

where Ψ 12 Ψ 12 ( r 1 , r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aaS baaSqaaiaaigdacaaIYaaabeaakiabggMi6kabfI6aznaaBaaaleaa caaIXaGaaGOmaaqabaGccaGGOaGaaCOCamaaBaaaleaacaaIXaaabe aakiaacYcacaWHYbWaaSbaaSqaaiaaikdaaeqaaOGaaiykaaaa@44A2@ , q=i q x +j q y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyCaiabg2 da9iaahMgacaWGXbWaaSbaaSqaaiaadIhaaeqaaOGaey4kaSIaaCOA aiaadghadaWgaaWcbaGaamyEaaqabaaaaa@3F07@ is two-dimensional vector.

Calculating the Fourier components in the Cartesian coordinate system, we obtain:

W 12q = π 2 i=1 n c i c j ζ ij 1 exp( q 12ij q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaaIXaGaaGOmaiaadghaaeqaaOGaeyypa0JaeqiWda3aaWba aSqabeaacaaIYaaaaOWaaabCaeaacaaMc8Uaam4yamaaBaaaleaaca WGPbaabeaakiaadogadaWgaaWcbaGaamOAaaqabaGccqaH2oGEdaqh aaWcbaGaamyAaiaadQgaaeaacqGHsislcaaIXaaaaaqaaiaadMgacq GH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaciyzaiaacIhacaGG WbWaaeWaaeaacqGHsislcaWGXbWaaSbaaSqaaiaaigdacaaIYaGaam yAaiaadQgaaeqaaOGaamyCamaaCaaaleqabaGaaGOmaaaaaOGaayjk aiaawMcaaaaa@5A64@ ,

W 1q = π 2 i=1 n c i c j ζ ij 1 exp( q 1ij q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaaIXaGaamyCaaqabaGccqGH9aqpcqaHapaCdaahaaWcbeqa aiaaikdaaaGcdaaeWbqaaiaaykW7caWGJbWaaSbaaSqaaiaadMgaae qaaOGaam4yamaaBaaaleaacaWGQbaabeaakiabeA7a6naaDaaaleaa caWGPbGaamOAaaqaaiabgkHiTiaaigdaaaaabaGaamyAaiabg2da9i aaigdaaeaacaWGUbaaniabggHiLdGcciGGLbGaaiiEaiaacchadaqa daqaaiabgkHiTiaadghadaWgaaWcbaGaaGymaiaadMgacaWGQbaabe aakiaadghadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaaa@5864@ ,

W 2q = π 2 i=1 n c i c j ζ ij 1 exp( q 2ij q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaaIYaGaamyCaaqabaGccqGH9aqpcqaHapaCdaahaaWcbeqa aiaaikdaaaGcdaaeWbqaaiaaykW7caWGJbWaaSbaaSqaaiaadMgaae qaaOGaam4yamaaBaaaleaacaWGQbaabeaakiabeA7a6naaDaaaleaa caWGPbGaamOAaaqaaiabgkHiTiaaigdaaaaabaGaamyAaiabg2da9i aaigdaaeaacaWGUbaaniabggHiLdGcciGGLbGaaiiEaiaacchadaqa daqaaiabgkHiTiaadghadaWgaaWcbaGaaGOmaiaadMgacaWGQbaabe aakiaadghadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaaa@5866@

where q 12ij =0.25 ς ij 1 ( α ij +2 β ij + γ ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa aaleaacaaIXaGaaGOmaiaadMgacaWGQbaabeaakiabg2da9iaaicda caGGUaGaaGOmaiaaiwdacqaHcpGvdaqhaaWcbaGaamyAaiaadQgaae aacqGHsislcaaIXaaaaOWaaeWaaeaacqaHXoqydaWgaaWcbaGaamyA aiaadQgaaeqaaOGaey4kaSIaaGOmaiabek7aInaaBaaaleaacaWGPb GaamOAaaqabaGccqGHRaWkcqaHZoWzdaWgaaWcbaGaamyAaiaadQga aeqaaaGccaGLOaGaayzkaaaaaa@5376@ ,

q 1ij =0.25 ς ij 1 γ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa aaleaacaaIXaGaamyAaiaadQgaaeqaaOGaeyypa0JaaGimaiaac6ca caaIYaGaaGynaiabek8awnaaDaaaleaacaWGPbGaamOAaaqaaiabgk HiTiaaigdaaaGccqaHZoWzdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa @4741@ , q 2ij =0.25 ς ij 1 α ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa aaleaacaaIYaGaamyAaiaadQgaaeqaaOGaeyypa0JaaGimaiaac6ca caaIYaGaaGynaiabek8awnaaDaaaleaacaWGPbGaamOAaaqaaiabgk HiTiaaigdaaaGccqaHXoqydaWgaaWcbaGaamyAaiaadQgaaeqaaaaa @473A@ .

To calculate the integral expressions in potential energy, we use the relations:

Ψ 112 2 r 12 d σ 12 = W 12q 2πq dq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada WcaaqaaiabfI6aznaaDaaaleaacaaIXaGaaGymaiaaikdaaeaacaaI YaaaaaGcbaGaamOCamaaBaaaleaacaaIXaGaaGOmaaqabaaaaaqabe qaniabgUIiYdGccaWGKbGaaC4WdmaaBaaaleaacaaIXaGaaGOmaaqa baGccqGH9aqpdaWdbaqaamaalaaabaGaam4vamaaBaaaleaacaaIXa GaaGOmaiaadghaaeqaaaGcbaGaaGOmaiabec8aWjaaykW7caWGXbaa aiaadsgacaWHXbaaleqabeqdcqGHRiI8aaaa@5129@ ,

Ψ 12 2 r 1 d σ 12 = W 1q 2πq dq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada WcaaqaaiabfI6aznaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaaa keaacaWGYbWaaSbaaSqaaiaaigdaaeqaaaaaaeqabeqdcqGHRiI8aO Gaamizaiaaho8adaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Za a8qaaeaadaWcaaqaaiaadEfadaWgaaWcbaGaaGymaiaadghaaeqaaa GcbaGaaGOmaiabec8aWjaaykW7caWGXbaaaiaadsgacaWHXbaaleqa beqdcqGHRiI8aaaa@4EF6@ ,

Ψ 12 2 r 2 d σ 12 = W 2q 2πq dq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipyI8Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbeqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada WcaaqaaiabfI6aznaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaaa keaacaWGYbWaaSbaaSqaaiaaikdaaeqaaaaaaeqabeqdcqGHRiI8aO Gaamizaiaaho8adaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Za a8qaaeaadaWcaaqaaiaadEfadaWgaaWcbaGaaGOmaiaadghaaeqaaa GcbaGaaGOmaiabec8aWjaaykW7caWGXbaaaiaadsgacaWHXbaaleqa beqdcqGHRiI8aaaa@4EF8@ .

Calculating the integrals over the wave vector in the polar coordinate system, we obtain for the Coulomb part of the energy functional of two-electron systems in 2D space:

V 2q = ( 1 r 12 Z( 1 r 1 + 1 r 2 ) ) Ψ 12 2 ( r 1 , r 2 )d σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIYaGaamyCaaqabaGccqGH9aqpdaWdbaqaamaabmaabaWa aSaaaeaacaaIXaaabaGaamOCamaaBaaaleaacaaIXaGaaGOmaaqaba aaaOGaeyOeI0IaamOwamaabmaabaWaaSaaaeaacaaIXaaabaGaamOC amaaBaaaleaacaaIXaaabeaaaaGccqGHRaWkdaWcaaqaaiaaigdaae aacaWGYbWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaGa ayjkaiaawMcaaiabfI6aznaaDaaaleaacaaIXaGaaGOmaaqaaiaaik daaaGccaGGOaGaamOCamaaBaaaleaacaaIXaaabeaakiaacYcacaWG YbWaaSbaaSqaaiaaikdaaeqaaOGaaiykaiaadsgacaWHdpWaaSbaaS qaaiaaigdacaaIYaaabeaaaeqabeqdcqGHRiI8aaaa@57F4@ ,

V 2q = π 2 π i,j=1 n c i c j 1 ς ij ( 1 α ij +2 β ij + γ ij Z( 1 α ij + 1 γ ij ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIYaGaamyCaaqabaGccqGH9aqpcqaHapaCdaahaaWcbeqa aiaaikdaaaGcdaGcaaqaaiabec8aWbWcbeaakmaaqahabaGaam4yam aaBaaaleaacaWGPbaabeaakiaadogadaWgaaWcbaGaamOAaaqabaGc daWcaaqaaiaaigdaaeaadaGcaaqaaiabek8awnaaBaaaleaacaWGPb GaamOAaaqabaaabeaaaaGcdaqadaqaamaalaaabaGaaGymaaqaamaa kaaabaGaeqySde2aaSbaaSqaaiaadMgacaWGQbaabeaakiabgUcaRi aaikdacqaHYoGydaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIa eq4SdC2aaSbaaSqaaiaadMgacaWGQbaabeaaaeqaaaaakiabgkHiTi aadQfadaqadaqaamaalaaabaGaaGymaaqaamaakaaabaGaeqySde2a aSbaaSqaaiaadMgacaWGQbaabeaaaeqaaaaakiabgUcaRmaalaaaba GaaGymaaqaamaakaaabaGaeq4SdC2aaSbaaSqaaiaadMgacaWGQbaa beaaaeqaaaaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaWcbaGaam yAaiaacYcacaWGQbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoa aaa@6BDE@ .

The energy of a two-electron system in 2D space is determined by the expression:

E 2 = min Ψ 12 ( T 2 + V 2q N 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIYaaabeaakiabg2da9maaxababaGaciyBaiaacMgacaGG UbaaleaacqqHOoqwdaWgaaadbaGaaGymaiaaikdaaeqaaaWcbeaakm aabmaabaWaaSaaaeaacaWGubWaaSbaaSqaaiaaikdaaeqaaOGaey4k aSIaamOvamaaBaaaleaacaaIYaGaamyCaaqabaaakeaacaWGobWaaS baaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaaa@47DA@

Results of variational calculations

Variation calculations were performed by random search methods and the Hook-Jeeves method. For a multiparameter system, the method of random search is the most accepted, because it does not lead to a long-term hang of the program associated with the search for a minimum of other methods. However, to increase the accuracy of the calculations, we used both methods. Table shows the results of the variational calculations of the energies of the negative 2D hydrogen ion and 2D helium atom in the singlet and triplet state. The calculated energy values are obtained for 30 independent exponents (n0=30) in the WF.4 The zero energy of the triplet term illustrates, that in the absence of a magnetic field, the 2D ion Hhas a single bound state that corresponds to the singlet term just like 3D ion H.11 For the D center, the energy of the exchange interaction, which can be defined as the difference between the energies of the singlet and the lowest triplet states, is exactly equal to the energy of the ground state. As can be seen from Table 1, for the triplet state of 2D H, the energy of the Coulomb repulsion of electrons tends to zero. With a further increase in the number of exponents in the WF (4), this assumption can be verified numerically.  We can obtain the useful boundary transitions from the two-electron system to the 2D H by equating to zero the last term in the Hamiltonian (1), corresponding to the interelectron repulsion. After performing variational calculations, we obtain for the case Vee = 0 (where Vee is the energy of interelectron interaction): in the singlet state, the energy H is equal to twice the energy of a two-dimensional hydrogen atom, which in atomic units is -3.999999. In the triplet state, the energy calculated by us is equal to -2.222219 at. un. The value of the energy of the triplet term obtained by us corresponds to the exact value of the total energy of the ground (-2.0 at. un.) and the first excited state (-0.(2) at. un.) of a 2D hydrogen atom.2 This example is a numerical illustration of the Pauli principle. Thus, electrons with opposite spins occupy the ground state, and when the spins are the same, one of the electrons necessarily passes into the next orbital with higher energy. For comparison, we present the value of the ground state energy of a two-dimensional hydrogen atom E1=-0.999995, obtained using the one-electron WF (2) for n = 14.

S

Z

E2

E2 [8]

E2 [9]

E2 [10]

T

Eext

Vee

V ee3D

0

1

-2.23938

-2.2338

-1.996

-1.96

2.239764

-5.66106

1.181455

0.3146

1

-1.99998

1.999978

-4.00139

0.001435

0

0

2

-11.8981

-11.8881

-11.6472

-11.56

11.8978

-27.3029

3.507033

0.9487

11.8904[8]

-27.2769[8]

3.4985[8]

1

-8.2816

-8.22

8.2718

-17.029

0.4753

Table 1 Total energy of singlet (S=0) and triplet (S=1) terms E2 (in at. un.) and various contributions to the energy of a 2D H (Z=1) and 2D He (Z=2) atoms: T is kinetic energy, Eexr is the total energy of electron interaction with the nucleus, Vee is the electron repulsion energy,  is the energy of the interelectron repulsion for 3D atoms

Discussion

This research presents an example of variational calculation of energies for two-electron atomic systems in two dimensions using Gaussian functions with correlation factors. The energy functional of two-electron 2D systems is presented in an analytical form and can be used to calculate both singlet and triplet terms of two-electron atoms with an arbitrary nuclear charge Z. The information about the spin state of the system is detected by automatically symmetrized WF (5). Singh8 proposed WFs that take into account the cusp conditions arising due to the Coulombic nature of external potential and electron–electron interaction, and the screening effects:

Ψ 2 2D ( r 1 , r 2 , r 12 )= C N exp(2Z r 1 )exp(2Z r 2 ) f 1 (a, r 1 , r 2 ) f 2 (b, r 12 ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aa0 baaSqaaiaaikdaaeaacaaIYaGaamiraaaakmaabmaabaGaamOCamaa BaaaleaacaaIXaaabeaakiaacYcacaWGYbWaaSbaaSqaaiaaikdaae qaaOGaaiilaiaadkhadaWgaaWcbaGaaGymaiaaikdaaeqaaaGccaGL OaGaayzkaaGaeyypa0Jaam4qamaaBaaaleaacaWGobaabeaakiGacw gacaGG4bGaaiiCaiaacIcacqGHsislcaaIYaGaamOwaiaadkhadaWg aaWcbaGaaGymaaqabaGccaGGPaGaciyzaiaacIhacaGGWbGaaiikai abgkHiTiaaikdacaWGAbGaamOCamaaBaaaleaacaaIYaaabeaakiaa cMcacaWGMbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadggacaGGSa GaamOCamaaBaaaleaacaaIXaaabeaakiaacYcacaWGYbWaaSbaaSqa aiaaikdaaeqaaOGaaiykaiaadAgadaWgaaWcbaGaaGOmaaqabaGcca GGOaGaamOyaiaacYcacaWGYbWaaSbaaSqaaiaaigdacaaIYaaabeaa kiaacMcacaWLa8Uaaiilaaaa@6A71@  (6)

where CNnormalization factor, a, b – variational parameters; 
f 1 (a, r 1 , r 2 )=cosh(a r 1 )+cosh(a r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaakiaacIcacaWGHbGaaiilaiaadkhadaWgaaWc baGaaGymaaqabaGccaGGSaGaamOCamaaBaaaleaacaaIYaaabeaaki aacMcacqGH9aqpciGGJbGaai4BaiaacohacaGGObGaaiikaiaadgga caWGYbWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabgUcaRiGacogaca GGVbGaai4CaiaacIgacaGGOaGaamyyaiaadkhadaWgaaWcbaGaaGOm aaqabaGccaGGPaaaaa@50F8@                             

f 2 (b, r 12 )=1+ r 12 exp(b r 12 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaakiaacIcacaWGIbGaaiilaiaadkhadaWgaaWc baGaaGymaiaaikdaaeqaaOGaaiykaiabg2da9iaaigdacqGHRaWkca WGYbWaaSbaaSqaaiaaigdacaaIYaaabeaakiGacwgacaGG4bGaaiiC aiaacIcacqGHsislcaWGIbGaamOCamaaBaaaleaacaaIXaGaaGOmaa qabaGccaGGPaaaaa@4B5B@

WF (6) is one of the variants of Slater orbitals with correlation factors.

As shown in Ref.8, these function give the lowest energies compared to calculations performed by other authors.9,10

We obtained the lowest values of the singlet state energy using Gaussian orbitals in comparison with the authors of Ref.8–10 Energy values of triplet terms are not given in Ref.8,10 Therefore, the Table shows only the values of the triplet states energy obtained in this research and in Ref.9 The advantage of Gaussian functions is related to the ability to reduce the finding of the energy functional minimum to the variation of analytic functions with many variables. A large number of variational parameters in a Gaussian system are not an obstacle for accurate results with using modern computers. Due to the existence of analogues of atomic systems with two electrons in condensed matter: two-electron centers of large radius in two-dimensional crystals ((bi)excitons, D centers), the computation methods described in this article can be used to calculate the energy spectrum of such systems. Gaussian functions with correlation factors have been successfully used for calculations the energies of atomic and molecular systems interacting with phonons in three-dimensional crystals.13,14

Conclusion

The variational method based on Gaussian functions was used to calculate the energies of singlet and triplet states of two-dimensional Hand He atoms. The received energies are the lowest in comparison with the ones obtained by other authors. The used method can be applied to calculate the energy of singlet and triplet states of the two-dimensional atoms in a magnetic field. In this case, the triplet state of H becomes stable. In magnetic fields, 2D atoms become anisotropic. This circumstance can easily be taken into account by the separation of components Gaussian WFs in the directions ОХ and OY. In this case, the anisotropy parameter should be introduced in the direction of the magnetic field in the 2D plane.

Acknowledgments

None.

Conflicts of interest

The author declares there is no conflict of interest.

Funding

None.

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