Two-dimensional confinement of hydrogen molecular ion

Accepted Manuscript Two-dimensional confinement of hydrogen molecular ion Germán Campoy-Güereña, Martín Molinar-Tabares PII: DOI: Reference:

S2210-271X(16)30504-7 http://dx.doi.org/10.1016/j.comptc.2016.12.018 COMPTC 2335

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Computational & Theoretical Chemistry

Received Date: Revised Date: Accepted Date:

11 December 2015 8 December 2016 12 December 2016

Please cite this article as: G. Campoy-Güereña, M. Molinar-Tabares, Two-dimensional confinement of hydrogen molecular ion, Computational & Theoretical Chemistry (2016), doi: http://dx.doi.org/10.1016/j.comptc. 2016.12.018

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Two-dimensional confinement of hydrogen molecular ion

Germán Campoy-GüereñaA, Martín Molinar-TabaresB

A

Departamento de Investigación Física, Universidad de Sonora, Av. Rosales y Blvd. Transversal, Hermosillo,

Sonora, México, 83190, P.O. Box 5-088, e-mail address: [email protected] B

Organismo de Cuenca Noroeste, Comisión Nacional del Agua, Cultura y Comonfort, Edif. México, Tercer Piso,

Hermosillo, Sonora, México, 83280, e-mail address: [email protected]

ABSTRACT Using the Born – Oppenheimer approximation, and considering the nuclei fixed at the foci, a study of the hydrogen molecular ion (H2+), confined strongly in two dimensions by ellipses of different size ξ0, is done. The Schrödinger equation is solved numerically in elliptic coordinates (ξ, η), applying the separation of variables method. The equations for ξ and η are solved following an iterative process, until the energy and separation constant become consistent with the size of the confining ellipse characterized by the parameter ξ0 and the internuclear distance R. The energies for the ion at its ground state and the equilibrium distance between the nuclei are obtained, for different values of ξ0, as well as the polarizability of the molecule.

Keywords: Schrödinger equation Born – Oppenheimer approximation, energies of the ion, equilibrium distance between nuclei, polarizability of the molecule.

Corresponding author; Germán Campoy-Güereña, [email protected]

Introduction There are many references for the study of the three dimensional confinement of the hydrogen molecular ion, where the Schrödinger equation at the Born – Oppenheimer approximation is solved. In these references, the authors work with the variational or the power series method used by Jaffe [1] to solve the Schrödinger equation for the free H2+. LeSar and Herschbach [2,3] used the variational method to calculate the energy, the vibrational frequency and the polarizability of H2+ under rigid confinement inside spheroids. Ley-Koo and Cruz [4] also studied the rigid confinement within spheroids, solving the Schrödinger equation with the separation variables method and following the procedure used by Jaffe, determining the size of the confining surfaces as a function of the energy of the ion at its ground state and some excited states. Vincke and Baye [5] worked with the variational method in the study of the interaction of the confined H2+ with a magnetic field. Gorecki and Byers-Brown [6] used the power series method of Jaffe to calculate the energy of the molecular ion and the pressure exerted on it by the walls of the spheroid.

When the motion of the ion nuclei is considered, the concept of center of mass is introduced, which allows us to separate the Hamiltonian of the molecule into two parts, one of them describing the motion of the center of mass. Following this procedure, Shertzer and Greene [7] improved the previously calculated value of the polarizability of the H2+.

For the case of two-dimensional systems, the literature gives references of the study of confined H atoms [8-11] and molecular systems, some of these can be found in the References and Notes section of this letter. Fabbri and Ferreira da Silva [12] calculated with a variational method, the ground state energy and the dissociation energy of the twodimensional H2 molecule, recalculating the corresponding values for a three-dimensional

case. Jia-Lin and Jia-Jiong [13] studied the hydrogen molecular ion in two dimensions; solving the Schrödinger equation, they found the energy and the equilibrium separation

of the nuclei. Peeters et al [14] obtained the ground state energy of two parallel twodimensional classical atoms for different interatomic distances; this system was supposed to be confined by parabolical potentials and repelling each other by a Coulombian interaction. Patil [15] studied the H2+ and the H2, considering them as free systems and obtaining numerically the ground and some excited states energy.

In this work, we show the results of our study of the two-dimensional confinement of H2+ inside an ellipse of size ξ0 with R the foci separation. These results were obtained from the numerical solution of the Schrödinger equation in elliptical coordinates, and all of them are shown in the paper, according to the following scheme. Firstly, the two-dimensional Schrödinger equation for the Hydrogen molecular ion, at the Born – Oppenheimer approximation is introduced, and the energies of H2+ confined by ellipses of different sizes ξ0 are found. After this, the polarizability of the H2+ is calculated. The method which we employ in our calculations was used in a previous paper of Molinar and Campoy [16], where the three dimensional confinement of H2+ was studied.

1. Solution of the Schrödinger equation in elliptic coordinates. The Schrödinger equation for this system in atomic units is:

 1 2 1 1 1         E r1 r2 R   2

(1)

Where the ri are the distances from the electron to each nucleus and R is the internuclear distance.

Knowing the relationship between Cartesian and elliptical coordinates:



r1  r2 r r ,  1 2 2a 2a

with r1 

c  z 2  x 2

(2)

and r2 

c  z 2  x 2 , being

a the focal distance and c the semi-

major axis of the ellipse; we can write the Laplacian operator in elliptical coordinates (ξ, η), and get the Schrödinger equation in the Born – Oppenheimer approximation for the hydrogen molecular ion in two dimensions:

 2 2 2  2  R 2   1    1     2 R   2   2 2 2 2  2 2   2 R       E  , 













  , 

(3)



where R = 2 a . Proposing a solution of the form: Ψ = (ξ) H(η) to start the separation of variables method, we get the following system of simultaneous equations

 2  d2 d R   1   2 R  RE  1 2    K   2 d 2 d  





(4)

And

  d2 d R 2   1    RE  1 2 H    KH   2 d 2 d  





(5)

where E is the energy eigenvalue and K is a separation constant which will act as a second eigenvalue. Calling M the differential operator in (5), we can approximate the eigenvalues Ki of M by expanding the solution H() in terms of the Legendre polynomials Pl():

(6)

 1 H     cl l  Pl   2 l 0

From Legendre’s equation:



 1  2

 dd

2 2

Pl    2

(7)

d Pl    l l  1Pl   d

and using their orthogonality, we can build a matrix

that satisfies:

 ~ Mcl  Kl cl

(8)

whose eigenvalues and eigenvectors are Kl and

M l ',l 

2l  12l '1 2

, and with elements

M l ',l given by

  d2 d d 2   P   1    2    2 Pl  d  l ' 2 1 d d d   1





which evaluating the integrals, becomes:

M l '.l

Int l / 2  2  l  l ',l  2l  1  2l  1  4k  l ',l  2 k k 1  1  l  1l  2 2l 2  2l  1 2l  1    l ',l  2   2l  12l  3 l ',l 2l  1  2l  3 2l  5  l l  1   l ',l  2  2l  1 2l  5 



(9)



where we have used the identity:



Int l / 2  d Pl    lPl     2l  1  4k Pl 2 k   d k 1

and other familiar recurrence relations of the Legendre Polynomials. We will consider a finite dimension NK of M in order to get the approximate eigenvalues Kl of this matrix; generally NK = 25 has proved to be good enough for our purpose. We then use the minimum eigenvalue K0 so obtained, to solve the eigenvalue equation (4) with a fixed value of R and starting values {K0, E0} and replace the minimum eigenvalue E1 so obtained, in equation (7); then using the values {K0, E1}, the eigenvalues of the matrix M are again calculated, selecting the minimum K1, starting from here an iteration process in which we replace alternately the minimum eigenvalue Ki obtained in (7) and the next minimum Ei calculated from (4) and so on, until a satisfactory convergence criteria is achieved.

Now, we will describe the iterative method with which we solved equation (4) in order to obtain the energy eigenvalues used in the above described iteration process. First we make the change of variable:

 

 1  1

(10)

in order to convert the working interval from [1, ∞) to [0, 1]. This turns (4) into

2  1  3 1   3 d  2 R 1   2  4 d    1     2 d d 2  R  2   RE  1  K  1      K  2  







(11)



Now, in order to calculate the energy eigenvalues Ei we use the following steps [16,17]:

(a) We consider as a function of both and E:  = E). (b) We define:

.

(c) We take the partial derivative of equation (11) with respect to E, obtaining:

3 2   1  3 1    d 4 d   2R 1   2   1    2 2 d d 





(12)

R 2 1    R  2   , E   0  RE  1  K  1    , E   2 2  





2

which together with eq. (10), constitute a pair of simultaneous equations which can be solved by the power series method for values Ki and Ei, obtaining a refined value Ei+1 through the Newton-Raphson formula:

Ei 1  Ei 

 0 , Ei    0 , Ei 

(13)

where 0 corresponds to 0, the value of  which defines the confining ellipse. The new value Ei+1 is replaced in equations (10) - (12) obtaining Ei+2; repeating the cycle (normally 3 times) until |Ei+(n+1) – Ei+n| is smaller than a defined tolerance . This Ei+(n+1) will be taken as our next eigenvalue Ei+1 in the above defined cycle until we get {Kn, En} which will be the final pair of values that satisfy simultaneously equations (4) and (5).

In order to find the value of R = Req corresponding to the minimum possible energy for a given confinement 0, we minimize the function

Ri 1  Ri 

 d  R   dE  R  Ri  d2   2 R   dE  R  Ri

using again Newton-Raphson:

(14) ,

but now approximating the first and second derivatives of R with respect to E by evaluating E at five different values of R separated by a small quantity , getting a result with an error of the order of 5. By using the described procedure, the energies for the ion ground state 1sg+ are obtained for R = 0.5 a.u. (Table 1 and Fig. 1).

TABLE 1

FIGURE 1

in its

According to Table 1 and Figure 1, we can see that if the size 0 of the confining ellipse increases enough, the energy tends to a fixed value (i.e., the molecule is practically free), this value corresponds to the energy of free ion with R = 0.5 a.u., -2.82157705 a.u., this value is slightly lower to the value cited in reference [15], -2.8213 a.u.. Observing Fig. 2,

we may appreciate that Req decreases as the size parameter ξ0 increases. About this figure it should be noted that as ξ0 is diminished the ellipse tends to be well defined, with a not reachable limiting value of 0 = 2 and with a corresponding eccentricity e = 1/2, but as ξ0 increases the ellipse evolve rapidly to take the appearance of a circle. Here we are faced with a remarkable physical restriction which prevents the equilibrium separation of the nuclei Req to reach the length of the semi-major axis a of the ellipse. This is an effect due to the pressure exerted by the wall of the confinement (the ellipse) on the ion. It must be said that if we try to find the equilibrium separation Req of the protons for a value of 0 slightly bigger than 2, say 2 +  then this separation increases as 1/ and so does the semi-major axis a , so the area of the ellipse increases accordingly.

FIGURE 2

Now, as we mentioned before, if the size 0 of the confining ellipse increases enough, we can see that the energy and the internuclear equilibrium distance tends to a fixed limiting values associated to the free ion, this procedure led us to find a free ion energy of 2.82237060 A.u., with an internuclear equilibrium separation Req of 0.51458032 a.u., these values can be compared with those cited in reference [13], -2.823 A.u. and 0.511 a.u. (see trend of energy on Table 2).

TABLE 2

The method that we are proposing allows to obtain the energy of the first excited state 2pu+ for any value of 0, this energy is shown in Table 2. It is interesting to observe how the energy of this state also approaches the value of -2.00 a.u. when 0 tends to 2.00 a.u., giving rise in this limit to a degenerated ground state.

2. Polarizability. The polarizability is approximated by perturbation theory to second order (E(2)) in the energy [2,16].

For a hydrogen molecule, ionized or not, the polarizability has two

components, the parallel

  and the perpendicular to the axis of the molecule (  ). As 

we are taking the coordinate system ZX such that the nuclei are placed on the Z axis, then we would have the following expressions for the components of the polarizability in the Kirkwood approximation [3]:

   zz  4 z 2 ,     xx  4 x 2

(15)

The components α║ and α┴ determine the mean polarizability (or simply polarizability) as





1    2



(16)

Table 3 shows the values of polarizability and Fig. 4 its behavior, considering different sizes ξ0 of the confining ellipse. The expected values used to calculate the components

and (   ) were found by numerical integration [18] of the density of probability (constructed with the series expansion of the eigenfunction), multiplied by z2 or x2.

TABLE 3

We can observe from Table 3 that as 0 approaches 2, the polarizability increases. This can be explained by the fact that in this limit the two protons reach a situation where their mutual repulsion is compensated by the pressure of the wall of the confinement and the electron moves practically free in the ellipse’s inside. On the other side, when 0 is very large the situation corresponds to a free ion and consequently the polarizability tends to a well defined limit.

FIGURE 3

Conclusions The energy of the two-dimensional confinement of H2+ has the expected behavior: energy decreases as the size of the confining ellipse ξ0 increases, and tends to a constant value for large values of ξ0, this constant value corresponds to the free ion energy.

As ξ0 increases, there is a displacement of the nuclei such that both particles approach to each other and the situation tends to that of the free ion. For the case of a free ion, an energy of -2.82237060 A.u. was found, with a separation of 0.51458031 a.u. between the nuclei.

If ξ0 approaches 2 from above, the internuclear distance grows indefinitely giving rise to an ellipse with an enormous inter-focal distance and area. This effect is due to the pressure exerted by the nuclei on the wall (perimeter) of the ellipse and is a very amazing and unique property that characterizes this system.

When the size of the ellipse 0 tends to 2.00 a.u. and R = Req, the energy of the first excited state 2pu+ approaches the state 1sg+ energy, giving rise to a degenerated ground state with an energy equal to four times that of the free bi-dimensional hydrogen atom [11].

References and Notes [1] G. Jaffe, Zur Theorie des Wasserstoffmoleküllions, Z. Physik 87 (1933) 535-542. [2] R. LeSar, D. R. Herschbach, Electronic and Vibrational Properties of Molecules at High Pressures, J. Phys. Chem. 85 (1981) 2798-2804. [3] R. LeSar, D. R. Herschbach, Polarizability and Quadrupole Moment of a Hydrogen Molecule in a Spheroidal Box, J. Phys. Chem. 87 (1983) 5202-5206. [4] E. Ley-Koo, S. A. Cruz, The hidrogen atom and the H2+ and HeH++ molecular ions inside prolate spheroidal boxes, J. Chem. Phys. 74 (1981) 4603-4610. [5] M. Vincke, D. Baye, Variational study of the hydrogen molecular ion in very strong magnetic fields, J. Phys. B: At. Mol. Phys. 18 (1985) 167-176. [6] J. Gorecki, W. Byers Brown, On the ground state of the hydrogen molecule-ion H2+ enclosed in hard and soft spherical boxes, J. Chem. Phys. 89 (1988) 2138-2148. [7] J. Shertzer, C. H. Greene, Nonadiabatic dipole polarizabilities of H2+ and D2+ ground states, Phys. Rev. A, 58 (1998) 1082-1086. [8] B. Zaslow and A. Kozycki, Two-dimensional analog to the hydrogen atom, Am. J. Phys. 35 (1979) 1118-1119.

[9] J. Wen-Kuang and M. E. Zandler, Hydrogen atom in two dimensions, Am. J. Phys. 47 (1979) 1005-1006. [10] G. Q. Hassoun, One- and two-dimensional hydrogen atoms, Am. J. Phys. 49 (1981) 143-146. [11] N. Aquino and E. Castaño, The confined two-dimensional hydrogen atom in the linear variational approach, Rev. Mex. Fis. E 51 (2005) 126-131. [12] M. Fabbri and A. Ferreira da Silva, Variational calculation of the two-dimensional H2 molecule, Phys. Rev. A 32 (1985) 1870-1872. [13] Jia-Lin Z. and Jia-Jiong X., Hydrogen molecular ions in two dimensions, Phys. Rev. B 41 (1990) 12 274-12 277. [14] F. M. Peeters, B. Partoens, V. A. Schweigert and G. Goldoni, Classical molecules in two dimensions, Phys. E 1 (1997) 219-225. [15] S. H. Patil, Hydrogen molecular ion and molecule in two dimensions, J. of Chem. Phys. 118 (2003) 2197-2205. [16] M. Molinar-Tabares, G. Campoy-Güereña, Hydrogen Molecular Ion Confined by a Prolate Spheroid, J. Comput. Theor. Nanosci. 9 (2012) 894-899. [17] G. Campoy, A. Palma, On the numerical solution of the Schrödinger equation with a polynomial potential, Int. J. of Quant. Chem. 20 (1986) 33-43. [18] Francis Scheid, Theory and Problems of Numerical Analysis, Schaum’s Outline Series, first ed., McGraw – Hill Book Company, New York, 1968.

FIGURE 1

Fig 1. Energy of H2+ with R = 0.5 a.u. confined in an ellipse, versus ξ0.

FIGURE 2

Fig.2. Confining ellipses for: a) 0 = 2.1, R = 2.04692, b) 0 = 4.0, R = 0.70007 and c) 0 = 8.0, R =0.51792; R in a.u. and dimensionless. The nuclei positions are illustrated at the ellipses foci on the Z axis.

FIGURE 3

Fig. 3. Polarizabilities versus ξ0.

TABLE 1 Table 1 Energy of the H2 ion under two-dimensional confinement, for several sizes ξ0 of the confining ellipse, with R = 0.5 a.u.

0

Energy [a.u.]

50.00

-2.82157705

5.00

-2.71461659

4.50

-2.61533853

4.00

-2.42225618

3.50

-2.03837900

3.00

-1.24304353

2.50

0.52927452

2.25

2.22201423

2.10

3.72441968

TABLE 2 Table 2 Minimal internuclear equilibrium distance and energy of H2+ at ground state 1sg+ and first excited state 2pu+ for some values of the confinement 0. 1sg+

2pu+

0

Req [a.u,]

Energy [a.u.]

Energy [a.u.]

50.00

0.51458032

-2.82237060

0.67238343

10.00

0.51483488

-2.82228594

0.72246882

5.00

0.59401353

-2.76081336

0.83815893

4.00

0.70007376

-2.65418571

0.06381161

3.50

0.79758790

-2.55658920

0.08452325

3.00

0.95905221

-2.41368287

-0.42543810

2.50

1.27542493

-2.21330728

-1.08187545

2.25

1.60895027

-2.09458741

-1.46980983

2.10

2.04691939

-2.02758242

-1.74477236

2.02

2.85175696

-2.00331904

-1.94001234

2.01

3.19786368

-2.00195295

-1.96887509

TABLE 3 Table 3 Calculated polarizability for some values of ξ0 (in a.u.).

0

2

2



2







2.01

2.55100

1.75119

4.30220

26.03059

12.26671

19.14865

2.02

1.98177

1.46533

3.44710

15.70957

8.58878

12.14918

2.10

0.92498

0.88650

1.81148

3.42233

3.14355

3.28294

2.50

0.33532

0.41824

0.75356

0.44977

0.69968

0.57473

3.00

0.22230

0.27698

0.49929

0.19768

0.30688

0.25228

5.00

0.13627

0.15721

0.29348

0.07428

0.09886

0.08657

8.00

0.13126

0.14540

0.27666

0.06891

0.08457

0.07674

50.00

0.13193

0.14568

0.27761

0.06962

0.08489

0.07726

Highlights.



The two-dimensional confinement of the H2+ with ellipses of size ξ0 is studied.



The energy and polarizability of the H2+ are calculated for different sizes of ξ0.



For different ξ0 the internuclear equilibrium distance between nuclei is found.