A CALCULATION OF SEMI-EMPIRICAL ONE-ELECTRON WAVE FUNCTIONS FOR MULTI-ELECTRON ATOMS USED FOR ELEMENTARY PROCESS SIMULATION IN NONLOCAL PLASMA

А -Е Е Е А Ы Е , Е А я варь–февра ь 2017 17№ 1 ISSN 2226-1494 http://ntv.ifmo.ru/

SCIENTIFIC AND TECHNICAL JOURNAL OF INFORMATION TECHNOLOGIES, MECHANICS AND OPTICS January–February 2017 Vol. 17 No 1 ISSN 2226-1494 http://ntv.ifmo.ru/en

537.5

А

А

А

А

А

А

А

А

А

. .

a

, . .

ё

b

,

А

. .

b

a

,

, 91405,

b

,

-

, 197101,

: alex_chirtsov@mail.ru

04.12.16, 30.12.16

doi: 10.17586/2226-1494-2017-17-1-159-171 –

: . ., ё . ., . .

//

- , . 2017. . 17. № 1. .159–171. doi:

10.17586/2226-1494-2017-17-1-159-171

А

.

,

.

,

,

,

.

.

.

,

.

,

.

.

.

.

.

.

.

.

.

,

.

,

,

,

,

,

,

,

,

A CALCULATION OF SEMI-EMPIRICAL ONE-ELECTRON WAVE FUNCTIONS

FOR MULTI-ELECTRON ATOMS USED FOR ELEMENTARY PROCESS

SIMULATION IN NONLOCAL PLASMA

M.V. Tchernycheva

a

, S.V. Sychov

b

, A.S. Chirtsov

b

a

Institut d'Electronique Fondamentale, Paris, 91405, France

b

ITMO University, Saint Petersburg, 197101, Russian Federation

Corresponding author: alex_chirtov@mail.ru

Article info

Article in Russian

For citation: Tchernycheva M.V., Sychov S.V., Chirtsov A.S. A calculation of semi-empirical one-electron wave functions for multi-electron atoms used for elementary process simulation in nonlocal plasma. Scientific and Technical Journal of Information Technologies,

Mechanics and Optics, 2017, vol. 17, no. 1, pp. 159–171. doi: 10.17586/2226-1494-2017-17-1-159-171

Abstract

Subject of Research.

The paper deals with development outcomes for creation method of one-electron wave functions of

complex atoms, relatively simple, symmetrical for all atom electrons and free from hard computations. The accuracy and

resource intensity of the approach are focused on systematic calculations of cross sections and rate constants of elementary

processes of inelastic collisions of atoms or molecules with electrons (ionization, excitation, excitation transfer, and others).

Method.

The method is based on a set of two iterative processes. At the first iteration step the Schrödinger equation was

solved numerically for the radial parts of the electron wave functions in the potential of the atomic core self-consistent field.

At the second iteration step the new approximation for the atomic core field is created that uses found solutions for all

one-electron wave functions. The solution optimization for described multiparameter problem is achieved by the use of genetic

algorithm. The suitability of the developed method was verified by comparing the calculation results with numerous data on

the energies of atoms in the ground and excited states.

Main Results

. We have created the run-time version of the program

for creation of sets of one-electron wave functions and calculation of the cross sections and constants of collisional transition

rates in the first Born approximation. The priori available information about binding energies of the electrons for any

many-particle system for creation of semi-empirical refined solutions for the one-electron wave functions can be considered at any

step of this procedure.

Practical Relevance

. The proposed solution enables a simple and rapid preparation of input data for

the numerical simulation of nonlocal gas discharge plasma. The approach is focused on the calculation of discharges in

complex gas mixtures requiring inclusion in the model of a large number of elementary collisional and radiation processes

involving heavy particles in different quantum states.

Keywords

multi-electron atoms, one-electron wave functions, self-consistent field, atomic core polarization, Born method, electron

impact ionization, electron impact excitation, optimization, genetic algorithms

(

1879 .,

«

» 1928 . [1])

[2–4].

-,

,

.

.

,

,

,

,

[5–9].

,

,

,

[10–13].

,

-.

(

.,

, [14]),

-,

,

[9].

,

.

,

.

[15–17],

[18],

[19]

,

.

(

)

.

,

.

[20, 21]

.

,

[22]

,

.

-,

,

.

-

,

,

-,

-.

-

,

.

,

--

-,

-

.

(

)

(

-

) ,

,

-,

.

-,

,

.

(

,

. .),

.

[23],

(

)

-.

,

«

-»

(

Δ

L

= ±1,

Δ

S

= 0)

-.

(

,

)

-,

«

» (

Δ

L

≠

±1)

(

Δ

S

≠

0)

[24, 25].

-,

,

(

,

,

,

-) [23].

σ

γ

0

γ′

σ

χ

:

0

2

B 2

γ γ 0 2 χ 3

χ '

8

σ

π

( )

(2

1)

k k

k k

dq

a

R q

k

l

q

 









_

_



 

, (1)

R

χ

-q

,

(

k

)

(

k

′

)

:

0

χ χ

0

χ

( )

(2

χ

1)(2

1)(2

1)

( )

( ) (

)

0

0

0

r

nl n l

l

l

R q

l

l

R

r R

 

r j qr dr















_{ }

_







, (2)

a0

–

,

r

,

-.

(1) (2)

k

k

′

–

;

R

nl

(

r

) –

(

n

l

)

;

j

χ

(

qr

) –

-χ

.

(«

-»)

«

»

-.

(1)–(2)

,

(«

-»)

[26].

.

-

[27]

,

[28–30].

-,

,

,

-.

-,

,

Z

< 12.

70-

[31],

(«

»)

,

-.

-,

.

(«

»)

-R

nl

(

r

)

2

2 2

2

(

1)

ζ

( )

2

2

nl

( )

0

d

d

l l

r

W

R

r

r dr

r

dr

r



_

_



_

_



_











, (3)

ζ

( )

r

r

–

;

W

–

.

,

,

n – l –

1

[32].

(3):

 

(

0) ~

,

(

) ~ exp

,

2 .

l

nl

nl

R r

r

R r

kr

k

W



 



 

,

-.

-,

:

 

( )

( ) exp

m

exp(

)

nl m

m

R

r



g r



kr





_

a r



_



kr







, (4)

(3),

Ϛ

(

r

) =

Z

,

a

m

g

(

r

)

-





0 , 0

1

δ

,

(

1)

2

,

1

2

(

1)

l

m m

a

k m

Z

a

a

m

m

l l









_{ }



_

_



_

_{ }

_



(5)

δ

l,0

–

-.

,

,

-)

g

(

r

)

m

=

n

– 1,

,

,

-.

-W

n

=

Z

/(2

n

2

)

-.

,

-,

,

n

*

= (–

Z

/2

W

)

1/2

.

(5),

k

*

=

Z

/

n

*

.

,

g

(

r

)

(5)

,

g

(

r

)

,

.

-,

(4)

l

=

n

–1,

[26].

,

,

-,

--

,

.

–

[33].

-,

.

(3)

,

,

.

-,

,

,

-

,

,

.

[31]

,

-,

,

-.

ω

,

-Ϛ

(

r/

ω

)

-.

ω

,

rmax

,

(3)

-.

. 1,

3

1 1

He I.

(

. 1, )

,

1s-

I.

3p-

,

ω

= 1,021,

,

(

. 1, ).

rmax

≈

5,8

a0

.

(

. 1, )

.

«

»

,

,

.

. 1. Э

3

1 1

:

( );

( );

( ).

,

–

q

,

.

. 2

1

r

,

.

.

1 2 3 4

1

–1

R

,

.

.

20 40 60

r

,

.

.

0,5

–0,5

R

,

.

.

10 20

.

[34].

(

)

-.

,

,

.

-.

-.

,

,

,

.

.

,

,

-.

,

-,

k

*j

= 1/

n

j*

(

n

j*

–

,

-)

,

-.

K

*

= {

k

*nl

}

R

= {

R

nl

(

r

)}.

p

,

-p

= (

n

,

l

)

a

p

,

R

:









2 2 2 2 ₂ 0 2 0

1

(

1)

2

( )

2

0,

( )

(

1)

( )

( ),

( )

( )

4

π

.

nlp p nlp nlp

nlp p p p p

p p

r

nl

nl

d

l l

rR

U

r

W

R

r dr

r

Z

U

r

a

r

a

r

r

R

r

r dr

r

dr

r

   









_





_











 







 







(6)

(6)

U

p

(

r

)

p

,

-,

,

φ

p

(

r

) –

,

-.

(6)

.

-N

= 200000

(

dr

= 0,015

).

:

,

2

)

1

(

)

(

2

)

(

)

(

2

)

(

,

)

1

(

)

1

(

)

(

,

)

1

(

)

1

(

)

(

2 2 2















_





























W

r

l

l

i

U

i

R

r

i

dR

i

R

d

dr

i

R

d

i

dR

i

dR

dr

i

dR

i

R

i

R

nl nl nl nl nl nl nl nl nl nl

(7)

P

nl

(

i

) –

;

dR

nl

(

i

)

d

2

R

nl

(

i

) –

.

R

nl

(0) =

δ

l0

,

dR

nl

(0) = 1,

d

2

R

nl

(0) = 0.

-U

nl

(

i

).

i

,

R

nl

(

j

)

.

R

nl

(

i

)

i

= 0, 1, 2.

W

nl

= – (1/2)/(

n

*

)

2

(

k

*nl

= 1/

n

*

)

-.

W

nl

(

)

(6),

-.

W

nl

,

,

(7), –

.

K

*

(

W

)

R

.

-,

.

-,

1 <

Z

< 10

2–8

,

0,5–5

.

,

-.

,

-.

,

W

p

c

p

= (

n

,

l

)

R

p

-,

,

(7).

{

R

p

(

r

i

)}.

W

p

=

W

p(j)

W

p

«

–

» (

W

p(j)

;

{

R

p

(

r

i

)}

(j)

)

,

.

-,

(

rmax

)

p

{

R

p

(

r

i

)}

(j)

,

W

p(j)

.

,

.

[35].

-.

.

-.

p

.

:

W

,

-,

.

-W1

–

W2

,

,

-,

,

.

-.

{

R

p

(

r

i

)}

(j)

.

rmax

(

k

nl

),

,

r

<

rmax

-R

nl

,

r

>

rmax

–

.

rmax

(

k

),

k



W

/13, 4 (

)

,

(

. 2).

. 2.

r

max

(

k

_nl

)

1,0



10

3

r

max

,

.

.

0,5



10

3

,

W

nl

-k

p

,

,

(7)

R

nl

.

,

,

,

(

)

rmax

(

k

p

),

C

i

,



3 2

 

3 2

 

2 1

 

2 1



3 1

/

/

i i i i i i i i

i

i i

r

r

k

k

r

r

k

k

C

k

k















,

k

j1

<

k

i2

<

k

i3

,

,

rmax

(

k

p

).

W

p

-k

p

,

.

-k

p

.

-,

-p

= (

n

,

l

).

k

p

-.



[

k k

i1

,

i2

,

k

i3

]



,

.

,

,

N

=100.

.

,

,

-C

i

< 0,

.

50–100

,

-k

.

,

C

j

< –2 ·10

6

.

C

i

k

,

10.

,

50

-,

,

-.

,

n+l

k

nli



[

k

nli

; ]



,

k

nli

.

15%,

,

-.

W

nlij

.

-.

.

Basic.

Intel(R) Core(TM) I5-2537M CPU 1.40GHz,

4 GB.

H He

-.

,

0,001,

H

50

Cu.

,

-,

.

.

U

(

r

)

n

l

,

,

U

(

r

),

n

,

l

,

W

.

nl

R

r

n

,

l

-. 3 4-.

-,

-.

-.

,

(5).

0,001,

.

(1)–(2)

,

20%.

-,

,

1,5–2

.

. 3.

2p-

(F I)

E

,

E

,

(

E

–

E

)/

E

,

%

H

13,6

13,54

0,44

He

24,6

24,8

–0,81

Li

5,4

4,77

11,67

Be

9,3

7,81

16,02

B

8,3

6,15

25,90

C

11,3

8,2

27,43

N

14,5

10,23

29,45

O

13,6

12,32

9,41

F

17,4

14,5

16,67

Na

5,1

4,47

12,35

.

1–3

.

-W

-,

.

-,

.

(

. 4)

(

)

.

,

.

«

»

,

.

–

-.

(

,

).

.

,

,

.

R

,

.

.

1

1,83

0 10 20

r

,

.

.

H(0) 2p

1

I

1s-1s

2

1s

1

2s

1

1s

1

2s

0

2p

1

1s

1

2s

0

2p

0

3s

1

1s

1

2s

0

2p

0

3s

0

3p

1

1s

1

2s

0

2p

0

3s

0

3p

1

3d

1

. 4.

I

(

)

(

)

R

,

.

.

0 10

r

,

.

.

H(0) 1s

1

He(0) 1s

2

R

,

.

.

0 10

r

,

.

.

H(0) 1s

1

He(0) 1s

2

R

,

.

.

0 10

r

,

.

.

H(0) 1s

1

He(0) 1s

2

R

,

.

.

0 10 20

r

,

.

.

H(0) 2s

1

He(0) 2s

2

R

,

.

.

0 10

r

,

.

.

H(0) 1s

1

He(0) 1s

2

R

,

.

.

0 10 20

r

,

.

.

H(0) 2p

1

He(0) 2p

1

R

,

.

.

0 10

r

,

.

.

H(0) 1s

1

He(0) 1s

2

R

,

.

.

0 10 20 30

r

,

.

.

H(0) 3s

1

He(0) 3s

1

R

,

.

.

0 10

r

,

.

.

H(0) 1s

1

He(0) 1s

2

R

,

.

.

0

r

,

.

. 46,31

H(0) 3p

1

He(0) 3p

1

R

,

.

.

0 10

r

,

.

.

H(0) 1s

1

He(0) 1s

2

R

,

.

.

0 10 20 30

r

,

.

.

H(0) 2d

1

,

-,

-,

-,

.

.

,

,

.

,

-,

,

-.

«

» [24].

,

,

:

,

(«

»)

,

.

( . .

)

,

-.

-,

,

.

1. Langmuir I. Oscillations in ionized gases // Proceedings of the National Academy of Sciences. 1928. V. 14. N 8. P. 627–637. doi: 10.1073/pnas.14.8.627

2. . . .

. .: , 1971. 490 .

3. . . . :

, 2009. 736 c.

4. . ., . ., . .

. .: , 2010. 512 .

5. Bogdanov E.A., Demidov V.I., Kaganovich I.D., Koepke M.E., Kudryavtsev A.A. Modeling a short DC discharge with thermionic cathode and auxiliary anode // Physics of Plasmas. 2013. V. 20. N 10. Art. 101605. doi: 10.1063/1.4823464 6. Koepke M.E., Walker J.J., Zimmerman M.I., Farrell W.M.,

Demidov V.I. Signature of gyro-phase drift // Journal of Plasma Physics. 2013. V. 79. N 6. P. 1099–1105. doi: 10.1017/S0022377813001128

7. Astafiev A.M., Gutsev S.A., Kudryavtsev A.A. Study of the discharge with an electrolytic electrode (Gatchina’s

discharge) // . 4. . .

2013. № 4. . 139–142.

8. Bogdanov E.A., Kudryavtsev A.A., Ochikova Z.S. Main scenarios of spatial distribution of charged and neutral components in SF6 plasma // IEEE Transactions on Plasma Science. 2013. V. 41. N 12. P. 3254–3267. doi: 10.1109/TPS.2013.2278839

9. . ., . ., . .,

. //

-, . 2016.

. 16. №5. . 903–916. doi: 10.17586/2226-1494-2016-16-5-903-916

10. Kaganovich I.D., Demidov V.I., Adams S.F., Raitses Y. Non-local collisionless and collisional electron transport in low-temperature plasma // Plasma Physics and Controlled Fusion. 2009. V. 51. N 12. Art. 124003. doi: 10.1088/0741-3335/51/12/124003

11. Tsendin L.D. Electron kinetics in non-uniform glow discharge

References

1. Langmuir I. Oscillations in ionized gases. Proceedings of the

National Academy of Sciences, 1928, vol. 14, no. 8, pp. 627–

637. doi: 10.1073/pnas.14.8.627

2. Granovskii V.L. Elektricheskii Tok v Gaze. Ustanovivshiisya Tok [Electric Current in the Gas. Sustained Current]. Moscow, Nauka Publ., 1971, 490 p.

3. Raizer Yu.P. Fizika Gazovogo Razryada [Gas Discharge Physics]. Dolgoprudnyi, Intellekt Publ., 2009, 736 p.

4. Kudryavtsev A.A., Smirnov A.S., Tsendin L.D. Fizika

Tleyushchego Razryada [Physics of Glow Discharge]. St.

Petersburg, Lan' Publ., 2010, 512 p.

5. Bogdanov E.A., Demidov V.I., Kaganovich I.D., Koepke M.E., Kudryavtsev A.A. Modeling a short DC discharge with thermionic cathode and auxiliary anode. Physics of Plasmas, 2013, vol. 20, no. 10, art. 101605. doi: 10.1063/1.4823464 6. Koepke M.E., Walker J.J., Zimmerman M.I., Farrell W.M.,

Demidov V.I. Signature of gyro-phase drift. Journal of

Plasma Physics, 2013, vol. 79, no. 6, pp. 1099–1105. doi:

10.1017/S0022377813001128

7. Astafiev A.M., Gutsev S.A., Kudryavtsev A.A. Study of the discharge with an electrolytic electrode (Gatchina’s discharge). Vestnik St. Petersburg State University. Ser. 4.

Phys. Chem, 2013, no. 4, pp. 139–142. (In Russian)

8. Bogdanov E.A., Kudryavtsev A.A., Ochikova Z.S. Main scenarios of spatial distribution of charged and neutral components in SF6 plasma. IEEE Transactions on Plasma

Science, 2013, vol. 41, no. 12, pp. 3254–3267. doi:

10.1109/TPS.2013.2278839

9. Tchernycheva M.V., Chirtsov A.S., Shvager D.A. Comparative analysis of plasma-chemical models for computer simulation of glow discharges in air mixtures.

Scientific and Technical Journal of Information Technologies,

Mechanics and Optics, 2016, vol. 16, no. 5, pp. 903–916. (In

Russian) doi: 10.17586/2226-1494-2016-16-5-903-916 10. Kaganovich I.D., Demidov V.I., Adams S.F., Raitses Y.

plasmas // Plasma Sources Science and Technology. 1995. V. 4. N 2. P. 200–211. doi: 10.1088/0963-0252/4/2/004

12. DeJoseph Jr., C.A., Demidov V.I., Kudryavtsev A.A. Nonlocal effects in a bounded low-temperature plasmas with fast electrons // Physics of Plasmas. 2007. V. 14. N 5. Art. 057101. doi: 10.1063/1.2436470

13. Walker J.J., Koepke M.E., Zimmerman M.I., Farrell W.M., Demidov V.I. Analytical model for gyro-phase drift arising from abrupt inhomogeneity // Journal of Plasma Physics. 2014. V. 80. N 3. P. 395–404. doi: 10.1017/S0022377813001359

14. Morgan (Kinema Research & Software) Database

[ ]. URL: www.lxcat.net/Morgan (

22.04.2016).

15. Tuhvatulin A.I., Sysolyatina E.V., Scheblyakov D.V. et. al. Non-thermal plasma causes P53-depended apoptosis in human colon carcinoma cells //Acta Naturae. 2012. V. 4. N 3. P. 82–87.

16. . ., . .

// . 2002. . 38. №6. . 43–49.

17. .

// . 2010. № 3. .

54–58.

18. . ., . ., . .,

. .

// . 2005. . 50. №2. . 361–366.

19. . ., . ., . .,

. ., . . ,

CES // .

2012. . 82. № 10. . 1–6.

20. Eliseev S.I., Kudryavtsev A.A., Liu H., Ning Z., Yu D., Chirtsov A.S. Transition from glow microdischarge to arc discharge with thermionic cathode in argon at atmospheric pressure // IEEE Transactions on Plasma Science. 2016. V. 44. N 11. P. 2536–2544. doi: 10.1109/tps.2016.2557587 21. Adams S.F., Demidov V.I., Bogdanov E.A., Koepke M.E.,

Kudryavtsev A.A., Kurlyandskaya I.P. Control of plasma properties in a short direct-current glow discharge with active boundaries // Physics of Plasmas. 2016. V. 23. N 10. P. 109901. doi: 10.1063/1.4941259

22. . . .

.: , 1963, 641 .

23. . . .: , 1967. 493 .

24. . ., . ., . .

. .: ,

1973. 142 .

25. . .

// . 1963. . 45. . 753–756.

26. . ., . .

// . - .

1970. . 51. . 90–123.

27. . :

. .: .

, 2006. 384 .

28. . . .:

, 1960. 256 .

29. . . . .: , 1976.

376 .

30. . . .: , 1979. . 2.

583 p.

31. . .

// . . 1961. . 15. . 3.

32. . ., . .

( ). 6- . .: , 2004.

. 3. 800 .

33. Parr R.G., Yang W. Density-Functional Theory of Atoms and Molecules. NY: Oxford University Press, 1989.

34. . ., . .

// .

4. . . 1991. № 1. . 146–149.

11. Tsendin L.D. Electron kinetics in non-uniform glow discharge plasmas. Plasma Sources Science and Technology, 1995, vol. 4, no. 2, pp. 200–211. doi: 10.1088/0963-0252/4/2/004 12. DeJoseph Jr., C.A., Demidov V.I., Kudryavtsev A.A.

Nonlocal effects in a bounded low-temperature plasmas with fast electrons. Physics of Plasmas, 2007, vol. 14, no. 5, art. 057101. doi: 10.1063/1.2436470

13. Walker J.J., Koepke M.E., Zimmerman M.I., Farrell W.M., Demidov V.I. Analytical model for gyro-phase drift arising from abrupt inhomogeneity. Journal of Plasma Physics, 2014, vol. 80, no. 3, pp. 395–404. doi: 10.1017/S0022377813001359

14. Morgan (Kinema Research & Software) Database. Available

at: www.lxcat.net/Morgan (accessed 22.04.2016).

15. Tuhvatulin A.I., Sysolyatina E.V., Scheblyakov D.V. et. al. Non-thermal plasma causes P53-depended apoptosis in human colon carcinoma cells.Acta Naturae, 2012, vol. 4, no. 3, pp. 82–87.

16. Stroikova I.K., Maksimov A.I. Disinfecting of solutions by glow and diaphragm discharges at atmospheric pressure.

Elektronnaya Obrabotka Materialov, 2002, vol. 38, no. 6, pp.

43–49. (In Russian)

17. Fromm V. Plasma surface treatment of contact lenses. Vestnik

Optometrii, 2010, no. 3, pp. 54–58. (In Russian)

18. Kalinin L.G., Panchenko G.I., Boshkova I.L., Kolomiichuk S.G. Influence of low-frequency and microwave electromagnetic fields on seeds. Biophysics, 2005, vol. 50, no. 2, pp. 334–337.

19. Kudryavtsev A.A., Chirtsov A.S., Yakovleva V.I., Mustafaev A.S., Tsyganov A.B. Electron energy spectra in helium observed in a microplasma collisional electron spectroscopy detector. Technical Physics. The Russian Journal of Applied

Physics, 2012, vol. 57, no. 10, pp. 1325–1330. doi:

10.1134/S1063784212100106

20. Eliseev S.I., Kudryavtsev A.A., Liu H., Ning Z., Yu D., Chirtsov A.S. Transition from glow microdischarge to arc discharge with thermionic cathode in argon at atmospheric pressure. IEEE Transactions on Plasma Science, 2016, vol. 44, no. 11, pp. 2536–2544. doi: 10.1109/tps.2016.2557587 21. Adams S.F., Demidov V.I., Bogdanov E.A., Koepke M.E.,

Kudryavtsev A.A., Kurlyandskaya I.P. Control of plasma properties in a short direct-current glow discharge with active boundaries. Physics of Plasmas, 2016, vol. 23, no. 10, pp. 109901. doi: 10.1063/1.4941259

22. Sobel'man I.I. Introduction to the Theory of Atomic Spectra. Moscow, Fizmatlit Publ., 1963, 641 p.

23. Born M. Atomic Physics. Moscow, Mir Publ., 1967, 493 p. (In Russian)

24. Vainshtein L.A., Sobel'man I.I., Yukov E.A. Secheniya

Vozbuzhdeniya Atomov i Ionov Elektronami [Excitation Cross

Sections of Atoms and Ions by Electrons]. Moscow, Nauka Publ., 1973, 142 p.

25. Ochkur V.I. On the method of Born-Oppenheimer approximation in the theory of atomic collisions.

Soviet Physics, 1963, vol. 45, pp. 753–756. (In Russian)

26. Vainshtein L.A., Presnyakov L.P. Calculation of the optically forbidden collision transitions in the second Born approximation. Trudy Fizicheskogo Instituta AN SSSR, 1970, vol. 51, pp. 90–123.

27. Mayer I. Theorems, Proofs, and Derivations in Quantum

Chemistry. NY, 2003.

28. Hartree D.R. The Calculation of Atomic Structures. NY: John Wiley and Sons, 1957.

29. Fok V.A. Nachala Kvantovoi Mekhaniki [Principles of Quantum Mechanics]. Moscow, Nauka Publ., 1976, 376 p. 30. Messiah A. Mécanique Quantique. Dunod, Paris, 1965. 31. Vainshtein L.A. Calculation of the wave functions and the

oscillator strengths complicated atoms. Trudy Fizicheskogo

Instituta AN SSSR, 1961, vol. 15, p. 3.

32. Landau L.D., Lifshits E.M. Quantum Mechanics

(Nonrelativistic Theory). 6th _{ed. Moscow, Fizmatlit Publ.,}

2004, vol. 3, 800 p.

33. Parr R.G., Yang W. Density-Functional Theory of Atoms and

Molecules. NY, Oxford University Press, 1989.

35. Sychov S., Chirtsov A. Genetic algorithm as a means for solving a radial Schrödinger equations system // Proc. 19th _Int.

Conf. on Soft Computing and Measurements. St. Petersburg, 2016. P. 265–267

group. Vestnik SPbSU. Seriya 4. Fizika. Khimiya, 1991, no. 1, pp. 146–149. (In Russian)

35. Sychov S., Chirtsov A. Genetic algorithm as a means for solving a radial Schrödinger equations system. Proc. 19th _Int.

Conf. on Soft Computing and Measurements. St. Petersburg,

2016, pp. 265–267

А

Authors

Ч ы а Ма я В а а –

-, ,

, 91405, , Maria.Tchernycheva@ief.u-psud.fr

Maria V. Tchernycheva – Researcher of the first class, Institut d'Electronique Fondamentale, Paris, 91405, France, Maria.Tchernycheva@ief.u-psud.fr

Сы ё С В а – , ,

- , 197101, ,

sergey.v.sychev@gmail.com

Sergey V. Sychov – postgraduate, ITMO University, Saint Petersburg, 197101, Russian Federation, sergey.v.sychev@gmail.com

Ч А а С – ,

, , , - ,

197101, , alex_chirtov@mail.ru