Simulation electronic thermes of two atoms molecules
SSimulation electronic thermes of two atoms molecules
Vladimir. P. Koshcheev , ∗ , Yuriy. N. Shtanov , ∗∗ Moscow Aviation Institute (National Research University),Strela Branch, Moscow oblast, Zhukovskii, 140180 Russia Tyumen Industrial University, Surgut Branch, Surgut, 628404 Russia
Abstract
In the first order of the perturbation theory, the correction to the electronic terms of a diatomic molecule iscalculated taking into account the Pauli principle.
Keywords: potential interaction energy, Pauli principle, Moli`ere approximation.
A new approach to calculating the potential energy of interaction of two atoms [1, 2] satisfactorily describesthe experimental results for atoms of noble gases, if the atomic form factor of an isolated atom is chosen in theMoli`ere approximation. In this publication, the electronic terms of the diatomic molecule (dimer) HF and HNewill be constructed in the first order of the perturbation theory.
The diatomic molecule will be described using the stationary Schr¨odinger equation Hψ = Eψ. (1)We represent the Hamiltonian of the equation (1) in the form H = H + U ; (2) U = Z Z e | r − r | + Z (cid:88) j =1 Z (cid:88) j =1 e | r + r j − r − r j | − Z (cid:88) j =1 Z e | r + r j − r | − Z (cid:88) j =1 Z e | r − r − r j | , where U –potential energy of interaction of two atoms; r and r – coordinates of the first and second atomicnucleus; r + r j and r + r j – coordinates of j -th and j -th electrons the first and second atom, respectively.The solution of the equation (1) with the Hamiltonian (2) will be sought using the perturbation theory ψ = ψ + ψ + . . . ∗ [email protected] ∗∗ [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b = E + E + . . . The electronic terms of a diatomic molecule will be sought in the first order of the perturbation theory E ( r ) = (cid:10) ψ (cid:12)(cid:12) U | ψ (cid:11) , (3)where r = | r − r | ; angle brackets (cid:104) . . . (cid:105) were introduced by Dirac [3].We represent the Hamiltonian H in the form H = H + H , where H i –the Hamiltonian of the i th atom; i =1,2.The solution to the Schr¨odinger equation H ψ = E ψ , will be sought in the form ψ = ψ ψ ,E = E + E , where the Schr¨odinger equation for the i th isolated atom has the form H i ψ i = E i ψ i , (4)where ψ i = ψ i ( r i , r i , . . . , r iZ i ).It is known [4] that using the variational principle from the stationary Schr¨odinger equation (4) one can constructthe Hartree-Fock equation. Hydrogen-like wave functions that approximate the solution of the Hartree-Fockequation for an isolated atom are presented in [5] ψ i = ϑ i ( r i ) ϑ i ( r i ) . . . ϑ iZ i ( r iZ i ) , (5)where ϑ ij = ϑ ij ( r ij )–hydrogen-like wave functions that form an orthonormal system.Using the formulas (2) and (5), we calculate (3) following [1] E ( r ) = (cid:90) E ( k ) exp ( i kr ) d k (2 π ) ; (6) E ( k ) = 4 πZ Z e k (cid:20) − F ( k ) Z (cid:21) (cid:20) − F ( k ) Z (cid:21) . The formula (6) does not take into account the Pauli principle between the electrons of the first and secondatoms. In [1], by analogy with (see, for example, [6]), it was proposed to take into account the Pauli principleusing the factor P ( k ) = (cid:20) − F ( k ) Z (cid:21) (cid:20) − F ( k ) Z (cid:21) . (7)The value F i ( k ) /Z i is the Fourier component of the atomic electron distribution plane, which is normalized tounity. As a result, we obtain an expression for the electronic term (potential energy of interaction of two atoms U ( r ) = E p ( r ) + . . . ) taking into account the Pauli principle in the form E p ( r ) = (cid:90) E ( k ) P ( k ) exp ( i kr ) d k (2 π ) . (8)The condition for the applicability of the correction in the first order of perturbation theory to the energy ofthe system in the unperturbed state has the form (cid:12)(cid:12) E p ( r ) (cid:12)(cid:12) (cid:28) (cid:12)(cid:12) E (cid:12)(cid:12) , where E = E + E ;the energies E i are also presented in [5] together with hydrogen-like wave functions thatapproximate the solution of the Hartree-Fock equation for an isolated atom.2 Calculation results and their discussion
The potential energy of interaction of two atoms was simulated when the form factor of the first atom waschosen in the Moli`ere approximation U ( r ) ≈ (cid:90) πZ Z e k (cid:20) − F ( k ) Z (cid:21) (cid:20) − F ( k ) Z (cid:21) exp ( i kr ) d k (2 π ) ; (9) F ( k ) = Z (cid:88) i =1 α i ( β i / a ) k + ( β i / a ) , (10)where α i , β i – Moli`ere approximation coefficients [7]; a ≈ . a Z − / ; a = 0 . F ( k ) was calculated using the wave function of the hydrogen atom in the 1 s -state F ( k ) = 16(4 + a k ) , (11)where a = a /Z ≈ . /Z – shielding length of a hydrogen atom.We calculate the expressions (9) using the formulas (10) and (11) U ( r ) ≈ e r (cid:104) exp (cid:16) − . ra (cid:17) (cid:16) . − . ra (cid:17) ++ exp (cid:16) − . ra (cid:17) (cid:16) − . − . ra (cid:17) ++ exp (cid:16) − . ra (cid:17) (cid:16) . − . ra (cid:17) ++ exp (cid:16) − ra (cid:17) (cid:16) . − . ra + 0 . r a − . r a (cid:17)(cid:105) . (12)Numerical calculations were performed using the program [8]. Figure 1 shows the graphs of the potential energyof interaction of atoms in the HF and HNe molecules. Since the atomic form factor for fluorine and neon atomswas chosen in the Moli`ere approximation, the results of calculating the potential interaction energy of atoms inHF and HNe molecules differ slightly from each other. It can be seen that the far potential minimum in Fig.1c,which corresponds to the area of action of van der Waals forces, is 10 ÷ times less deep than the near potentialminimum in Fig.1a.Figure 2 shows satisfactory agreement of the graph of the potential energy of interaction of atoms in the HFmolecule in comparison with the experimental data presented in [9]. In [2], satisfactory agreement with theexperiment was demonstrated for the plot of the potential energy of interaction of two neon atoms in the region ofaction of van der Waals forces. The form factors of neon atoms in [2] were taken in the Moli`ere approximation. Theagreement between the calculation results and experiment can be improved if the form factors of fluorine and neonatoms are constructed using hydrogen-like wave functions [5], which approximate the solution of the Hartree-Fockequation for an isolated atom. It can be seen that for the formation of the HNe molecule when hydrogen and neonatoms approach each other, it is necessary to overcome a potential barrier with a height of about 0 .
04 electronvolt,as can be seen in Fig. 1b. At low temperatures, the formation of HF and HNe dimers in the region of action ofvan der Waals forces is possible. The depth of the potential well is 10 − ÷ − electronvolts, and the distancebetween the atoms is 5 ÷ The reported study was funded by RFBR, project number 20-07-00236 a.
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Fig. 1 . Potential energy of interaction of atoms in molecules HF(solid line) and HNe (dashed line) at differentscales: a) r/a ∈ [0; 3], b) r/a ∈ [3; 8], c) r/a ∈ [8; 15]. Fig. 2 . The potential interaction energy U ( r ), which is calculated by formula (12) (solid line), in comparisonwith the experimental result, which is presented in [9] (circles), and the Morse potential [9] (dashed line) for theHF molecule. 5 U (r) , эВ r/a Figure 1a.
Potential energy of interaction of atoms in molecules HF(solid line) and HNe (dashed line) at r/a ∈ [0; 3] U (r) , − эВ r/a Figure 1b.
Potential energy of interaction of atoms in molecules HF(solid line) and HNe (dashed line) at r/a ∈ [3; 8] U (r) , − эВ r/a Figure 1c.
Potential energy of interaction of atoms in molecules HF(solid line) and HNe (dashed line) at r/a ∈ [8; 15] U (r) , c m − r, Ang. Figure 2.
The potential interaction energy U ( r ), which is calculated by formula (12) (solid line), in comparisonwith the experimental result, which is presented in [9] (circles), and the Morse potential [9] (dashed line) for the HFmolecule.), which is calculated by formula (12) (solid line), in comparisonwith the experimental result, which is presented in [9] (circles), and the Morse potential [9] (dashed line) for the HFmolecule.