Variational perturbation treatment of the confined hydrogen atom

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Variational perturbation treatment of the confined hydrogen atom

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IOP PUBLISHING

EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 32 (2011) 1275–1284

doi:10.1088/0143-0807/32/5/015

Variational perturbation treatment of the confined hydrogen atom H E Montgomery Jr Chemistry Program, Centre College, 600 West Walnut Street, Danville, KY 40422-1394, USA E-mail: [email protected]

Received 18 May 2011, in final form 6 June 2011 Published 8 July 2011 Online at stacks.iop.org/EJP/32/1275 Abstract

The Schrödinger equation for the ground state of a hydrogen atom confined at the centre of an impenetrable cavity is treated using variational perturbation theory. Energies calculated from variational perturbation theory are comparable in accuracy to the results from a direct numerical solution. The goal of this exercise is to introduce the student to the effects of confinement on atomic systems using a tractable problem from which insight into variational perturbation theory may be gained.

1. Introduction

Introductory quantum theory courses typically discuss the Schrödinger equation for a particle confined in a box with impenetrable walls and for the hydrogen atom. The union of these two problems, the hydrogen atom confined in an impenetrable spherical box with the proton in the centre, is not very well known. This problem was first studied over 70 years ago by Michels et al [1], who used it to model the effects of pressure on an atom’s energy and polarizability. It has subsequently been applied to a wide range of problems. The interested reader is referred to Sen et al [2] and references therein. With the development of the technology to construct atomic scale confinements, the study of confined systems has become increasingly relevant. In this work, energies were calculated over a range of confinement radii using variational perturbation theory. To demonstrate how variational perturbation theory facilitates the calculation of high-order energy corrections, the calculation was carried out through fifth order in the energy. The zero-order system was obtained by partitioning the Hamiltonian using a method developed by Sternheimer [3]. The variational perturbation energies are shown to be accurate by comparison with results from accurate numerical calculations and with exact results at selected confinement radii. This calculation could be useful as a computational quantum mechanics exercise or as a seminar presentation. Although this work is restricted to the 1s ground state, the extension to excited states is straightforward and could be undertaken as a computational laboratory problem or an exam problem. c 2011 IOP Publishing Ltd Printed in the UK & the USA 0143-0807/11/051275+10$33.00

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H E Montgomery

2. Variational perturbation theory

The derivation of Rayleigh–Schrödinger perturbation theory can be found in any introductory quantum theory text [4]. Only those parts required to develop variational perturbation theory are repeated here. The central idea is that for a system described by a Hamiltonian Hˆ , ground state eigenfunction and ground state eigenvalue E, a system can be found with a Hamiltonian Hˆ 0 , ground state eigenfunction ϕ0 and ground state eigenvalue ε0 that satisfies the same symmetry and boundary conditions as Hˆ . We thus require Hˆ 0 ϕ0 = ε0 ϕ0 .

(1)

Hˆ 0 is called the zero-order Hamiltonian, and the Hamiltonian can now be written as Hˆ = Hˆ 0 + λ Hˆ 1 ,

(2)

where Hˆ 1 is the first-order Hamiltonian and λ ∈ [0,1] is called the perturbation parameter. Mathematically, λ can be viewed as a continuous variable that, by increasing from 0 to 1, smoothly transforms Hˆ 0 into Hˆ . If no physically natural choice for λ exists, it may be used as an ordering parameter and then set equal to 1 at the end of the calculation. The Schrödinger equation is written as Hˆ = (Hˆ 0 + λ Hˆ 1 ) = E .

(3)

Since Hˆ depends on λ, its eigenfunctions and eigenvalues must also depend on λ. They can be expanded as power series in λ, = ψ0 + λψ1 + λ2 ψ2 + λ3 ψ3 + · · · ,

(4)

E = E0 + λE1 + λ2 E2 + λ3 E3 + · · · ,

(5)

and we set ψ0 = ϕ0 and E0 = ε0 . ψn is the nth-order correction to the ground state wavefunction, while En is the nth-order correction to the energy. Substituting into the Schrödinger equation and rearranging, we obtain (Hˆ 0 + λHˆ 1 − E0 − λE1 − λ2 E2 − λ3 E3 + · · ·)(ψ0 + λψ1 + λ2 ψ2 + λ3 ψ3 + · · ·) = 0.

(6)

Multiplying equation (6) and collecting the coefficients of like powers of λ gives a power series in λ. The only way that a power series can equal zero for all values of λ is for the coefficient of each power of λ to be equal to zero. This results in the Rayleigh–Schrödinger perturbation equations. The coefficients of λ0 and λ1 are (Hˆ 0 − E0 ) ψ0 = 0,

(7)

(Hˆ 0 − E0 ) ψ1 + (Hˆ 1 − E1 ) ψ0 = 0.

(8)

and

Equations (7) and (8) give ψ0 Hˆ 0 ψ0 dτ E0 = , ψ0 ψ0 dτ

(9)

ψ0 Hˆ 1 ψ0 dτ E1 = . ψ0 ψ0 dτ

(10)

The Rayleigh–Ritz variational principle [5] U | Hˆ − E |U 0,

(11)


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is used to obtain approximate corrections to the energy and wavefunction. The variational wavefunction, U, is expanded as a power series in λ, analogous to equation (4), as U = ϕ0 + λϕ1 + λ2 ϕ2 + λ3 ϕ3 + · · · ,

(12)

where for non-degenerate energy levels, one may write ψ0 = ϕ0 due to equation (1). It is important to note that ϕ1 , ϕ2 , ϕ3 , . . . are approximate wavefunctions. We define the notation ˆ n dτ, m| |n = ϕm∗ ϕ (13) ˆ is any operator (including unity). In the discussion to follow, we restrict our where consideration to real ϕn . Inserting the expansions for Hˆ , U and E into equation (11), we obtain 0|Hˆ 0 − E0 |0 + {21|Hˆ 0 − E0 |0 + 0|Hˆ 1 − E1 |0}λ + {21|Hˆ 0 − E0 |0 + 1|Hˆ 0 − E0 |1 + 21|Hˆ 1 − E1 |0 − E2 0|0}λ2 20|Hˆ 0 − E0 |3 + 20|Hˆ 1 − E1 |2 + 21|Hˆ 0 − E0 |2 λ3 + · · · 0 + + 21|Hˆ 1 − E1 |1 − E2 0|1 − E3 0|0 (14) and in general,

λm+n m| Hˆ 0 + λHˆ 1 |n −

m=0 n=0

λl El

l=0

λm+n m|n 0.

(15)

m=0 n=0

Expanding equation (15) gives a power series in λ. From equations (9) and (10), the coefficients of λ0 and λ1 are zero (due to the choice ϕ 0 as the unperturbed function in equation (1)) and the term involving λ2 must be the dominant term in the series. The term 1|Hˆ 0 − E0 |0 in the coefficient of λ2 is zero using equation (7) and (16) E2 1|Hˆ 0 − E0 |1 + 21|Hˆ 1 − E1 |0 provides a variational upper bound to E2 , the second-order energy and an approximate firstorder wavefunction, ϕ1 . Equation (16) is the variational perturbation equation developed by Hylleraas [6] in 1930. As noted by Scherr and Knight [7], equation (16) is the variational integral of the Euler equation for equation (8). To find E2 and ϕ1 , ϕ1 is expanded in a set of k trial functions, χi , as ϕ1 =

k

xi χi ,

(17)

i=1

where the xi are the variational parameters and the χi must satisfy the same symmetry and boundary conditions as Hˆ . It is helpful to define the two integrals PBi = χi | Hˆ 1 − E1 |ψ0 , (18)

(19) Ti,j = χi | Hˆ 0 − E0 χj . When the expansion for ϕ1 is inserted into equation (16), we obtain E2 2

k

xi PBi +

i=1

k k

xi xj Ti,j .

(20)

i=1 j =1

The xi s are determined by minimizing E2 with respect to each of the xi s. This gives a set of k simultaneous linear equations that can be written as T x = −2P B,

(21)

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where T is a k × k symmetric matrix and PB is a k-element column vector. Inversion of T to find T−1 gives x = −2T −1 P B.

(22)

Thus, the wavefunction coefficients can be obtained by one call to the matrix inverse function of a computer algebra system (CAS) followed by a second call to a matrix multiplication function to multiply T−1 and PB. For example, both Maple and Mathematica have linear algebra packages that include Gaussian elimination and linear solver functions that can solve equation (21) with a single function call. Hylleraas [6] developed an efficient procedure for solving equation (20) subject to the requirement that E2 be a minimum. An outline of his method, which is similar to the Gauss exclusion algorithm for matrix inversion, is provided in appendix A while a numerical example of its application to the confined hydrogen atom is presented in appendix B. Scherr and Knight [7] extended Hylleraas’ work by showing that, to the extent that E2 and ϕ1 are given accurately by equation (16), the coefficient of λ2 vanishes and the coefficient of λ3 provides an estimate of E3. An upper bound to E4 is given by the coefficient of λ4 and in general En is given by the coefficient of λn . In each case, the Rayleigh–Schrödinger perturbation equations for lower-order terms can be used to simplify the equation of interest. For the present calculation, we need the expressions for E3 , E4 and E5 , which are as follows: ∼ 1| H1 − E1 |1 − 2E2 1|0 , (23) E3 = E4 2| H0 − E0 |2 + 2 2| H1 − E1 |1 − 2 E2 2|0 − E2 1|1 − 2E3 1|0 ,

(24)

E5 ∼ = 2| H1 − E1 |2 − 2E2 2|1 − E3 1|1 − 2E3 2|0 − 2E4 1|0 .

(25)

Note that the knowledge of ϕn allows the calculation of E2n and E2n+1 . Equation (23) and ϕ1 are used to calculate E3 while equations (24) and (25) are used with ϕ2 to calculate E4 and E5 . To calculate E4 , ϕ2 is expanded in the form of equation (17), the PBi integrals are formulated as PBi = χi | H1 − E1 |ϕ1 − E2 χi | ψ0 ,

(26)

and Hylleraas’ method is used to evaluate equation (24). E5 then follows from equation (25). This process can be continued to whatever order is desired. The ϕi calculated by the Hylleraas method do not satisfy the requirement that the total wavefunction be normalized. Since the lack of normalization does not affect the calculation of the En, we will not calculate the normalized wavefunction. The interested reader is referred to section III of [7] for a detailed discussion of normalization. A fundamental difficulty of perturbation theory is finding an appropriate zero-order system. For a zero-order wavefunction ψ0 , Sternheimer [3] defined the zero-order potential (Tˆ ψ0 ) Vˆ 0 = E0 − , (27) ψ0 where Tˆ is the kinetic energy operator for the system and E0 is an arbitrary constant chosen to simplify the potential. The zero-order Hamiltonian Hˆ 0 is given by Hˆ 0 = Tˆ + Vˆ 0 . (28) For a Hamiltonian Hˆ , the perturbation potential Hˆ 1 is given by Hˆ 1 = Hˆ − Hˆ 0 .

(29) ˆ ˆ Knowing H0 , H1 and ψ0 , variational perturbation theory provides a method for calculating corrections to the wavefunction and energy.


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3. Confined hydrogen

The Hamiltonian in atomic units1 for a hydrogen atom is given by ˆ2 1 1 L 1 2 Hˆ = − ∂r (r ∂r ) − 2 − , 2 r2 r r ˆ 2 is the angular momentum operator such that where L ˆ 2 Y,m (θ, φ) = ( + 1) Y,m (θ, φ) , L

(30)

(31)

and Y,m (θ, φ) is the normalized spherical harmonic of section 14.30(i) of [8]. The radial boundary condition for a free hydrogen atom requires that the wavefunction exponentially approach zero as r → ∞. When the proton of the hydrogen atom is placed at the centre of an impenetrable sphere of radius rc , the radial boundary condition requires that ψ (rc ) = 0. Incorporating a cut-off function (rc − r) [9] in the wavefunction does this most easily. For the ground state zero-order wavefunction, we use ψ0 = N (rc − r) e−αr Y0,0 (θ, φ) ,

(32)

where α is a variational parameter. The zero-order potential is given by the Sternheimer method as − 2r12 ∂r (r 2 ∂r )[(rc − r) e−αr ] (rc − r) e−αr

1 α rc + 1 α α2 r E0 (rc − r) − + (4 + α rc ) − . (33) = rc − r r 2 2 2 If we choose E0 = − 2α + α2 , Vˆ 0 simplifies to rc

1 (α rc + 1) 2 α r Vˆ 0 = . (34) − + rc − r r rc With this choice of Vˆ 0 ,

(α rc + 1) 2αr 1 1 (35) − + Hˆ 0 = − 2 ∂r (r 2 ∂r ) + 2r rc − r r rc and

−1 2αr ˆ ˆ ˆ . (36) H1 = H − H0 = rc (1 − α) − 1 − r + r (rc − r) rc Since 0| H 0 + H1 |0 0| H |0 E0 + E1 = = , (37) 0 | 0 0 | 0 the Rayleigh–Ritz variational principle [5] can be used to find the value of α that minimizes E0 + E1 . The ϕn s were constructed from trial functions of the form Vˆ 0 = E0 −

χi = (rc − r) r i e−α r Y0,0 (θ, φ) . The PBi s and Ti,j s can be evaluated in terms of the integrals rc r n e−2α r d r = γ (n + 1, 2 α rc ) ,

(38)

(39)

0

where γ (α, z) is the incomplete gamma function as defined in section 8.2(i) of [8]. The Hartree, Eh, is the atomic unit of energy (1Eh = 4.359 743 81 × 10−18 J). The Bohr radius, a0, is the atomic unit of length (a0 =5.291 772 083 × 10−11 m).

1

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Figure 1. Energy of the confined hydrogen atom 1s ground state.

4. Results and discussion

The variational perturbation equations were programmed using the Maple 14 CAS. The use of a CAS facilitates the evaluation of the required integrals and finding the value of α. Figure 1 shows the 1s energy of a confined hydrogen atom from rc = 1 a0 to rc = 4 a0 . As rc increases, the effect of confinement decreases and the energy of the confined atom approaches the free atom value of −1/2 Eh . As rc decreases, the energy increases, passes through zero and rapidly rises to large values as the electron momentum increases. Since the electron is bound by the confinement, there is nothing inconsistent about a state with positive energy. When rc for a confined atom corresponds to the innermost node of the ns radial wavefunction of a free atom, the 1s wavefunction of the confined atom has the same energy as the ns wavefunction of the free atom. These exact energies provide a convenient check of the numerical accuracy of the variational perturbation procedure. A free atom wavefunction with principal quantum number n and angular quantum number has n − The 2s radial wavefunction √ − 1r nodes in the radial wavefunction. (2s) = 2 a0 . Thus, at rc = 2 a0 , the energy of R2s = (2 − r)/(2 2) e− 2 has a single node at rnode the 1s confined atom is −1/8Eh . √ r The 3s radial wavefunction for the free atom, R3s√= 2(27 − 8r + 2r 2 ) e− 3 /(81 3), has (3s) = 3(3 − 3)/2 = 1.901 924 a0 , so the confined two nodes, the innermost of which is at rnode atom energy is −1/18Eh at rc = 1.901 924 a0 . Similar nodes exist for larger values of n. Table 1 gives the energy corrections for the 1s state over a range of confinement radii. The calculations used a ten-term trial function and were carried out to fifth order in the energy. Although the first-order correction is large for small rc , the second- and higher-order energy corrections steadily decrease in magnitude. The sum of the variational perturbation energies through fifth order is in excellent agreement with the exact energies of Aquino et al [10] calculated by direct numerical solution of the Schrödinger equation, with the perturbation theory solutions of Laughlin [11] and Montgomery [12], and with the linear approximation of Djajaputra and Cooper [13].


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Table 1. En/Eh for the 1s state as a function of the confinement radius rc (in a0).

n

rc = 0.1/a0

rc = 1/a0

rc = 1.901 924/a0

rc = 2/a0

rc = 10/a0

0 1 2 3 4 5

−7.942 579 −0.971 509 −0.640 372 482.844 264 3.362 093 0.584 859 −6.445 057 −0.017 394 −0.000 042 0.583 194 0.000 840 −0.050 398 −0.000 041 0.003 840 0.000 002

−0.625 000 −0.564 794 0.500 000 0.064 893 −0.000 089 −0.000 007 −0.000 001

Total Exact [10]

468.993 264 468.993 039

2.373 991 −0.055 556 2.373 990 −0.555 556

−0.125 000 −0.499 999 −0.125 000 −0.499 999

5. Extensions

Some straightforward extensions of the above calculation are suggested for students who would like to develop some familiarity with computational quantum mechanics. Although this work only considered the 1s ground state, variational perturbation theory can be used to calculate the lowest excited states of a given symmetry, e.g. 2p and 3d. By writing ψ0 as ψ0 (n, ) = N (rc − r) r n e−αr Yn, (θ, φ) , (40) the Sternheimer method can be used to obtain Hˆ 0 and Hˆ 1 . The χi s take the form of equation (38) with Y0,0 (θ, φ) replaced by the appropriate Yn, (θ, φ). From the knowledge of the E (rc ) versus rc curves for both ground and excited states, the spectra for the confined atom can be calculated and the effects of changes in the confinement radius on those spectra can be studied. The force per unit area applied by a confined atom on its confining potential at confinement radius rc is called the electronic pressure [14] and is given by 1 ∂r E (rc ) . (41) P (rc ) = − 4π rc2 c ∂rc E (rc ) can be obtained by numerically differentiating the curve of E (rc ) versus rc at the confinement radius of interest. This provides an approximate model for the effects of pressure on the energy levels of an atom. While this paper has concentrated on applying variational perturbation theory to the confined hydrogen atom, the method can also be used to conveniently find second- and higher-order energy corrections to typical perturbation theory problems such as multipole polarizabilities and particles in infinite square wells with non-flat bottoms. Acknowledgment

The author acknowledges the reviewer whose recommendations significantly improved this work and L L Tankersley for a critical reading of the manuscript. Appendix A. Hylleraas’ method for calculation of E2

We wish to solve the equation E2 = 2

k i=1

xi ai +

k k i=1 j =1

xi xj Ai,j ,

where Ai,j = Aj,i ,

(A.1)

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subject to the condition that E2 be a minimum. We define 1 x1 = y1 − (A1,2 x2 + A1,3 x3 + A1,4 x4 + · · · + A1,k xk ). (A.2) A1,1 When this expression for x1 is inserted into equation (A.1), the x1 s are eliminated and equation (A.1) can be rewritten as k k k xi bi + xi xj Bi,j , (A.3) E2 = 2a1 y1 + A1,1 y12 + 2 i=2

i=2 j =2

where A1,i A1,j A1,i a1 and Bi,j = Ai,j − . A1,1 A1,1 The x2 s can be eliminated by defining 1 x2 = y2 − (B2,3 x3 + B2,4 x4 + B2,5 x5 + · · · + B2,k xk ), B2,2 which after substitution into equation (A.3) gives k k k xi ci + xi xj Ci,j , E2 = 2a1 y1 + a1,1 y12 + 2B2 y2 + B2,2 y22 + 2 bi = ai −

i=3

(A.4)

(A.5)

(A.6)

i=3 j =3

where B2,i B2,j B2,i b2 and Ci,j = Bi,j − . B2,2 B2,2 We can continue in this way through k steps and finally obtain E2 = 2a1 y1 + A1,1 y12 + 2b2 y2 + B2,2 y22 + 2c3 y3 + C3,3 y32 + 2d4 y4 + D4,4 y42 + · · · + 2kk yk + Kk,k yk2 . ci = bi −

(A.7)

(A.8) (A.8)

Minimizing E2 with respect to each of the yi s gives −a1 −b2 −kn y1 = , y2 = , . . . , yk = , (A.9) A1,1 B2,2 Kn,n which after substitution into equation (A.8) gives c2 k2 a2 b2 d2 (A.10) E2 = − 1 − 2 − 3 − 4 − · · · − k . A1,1 B2,2 C3,3 D4,4 Kk,k We now have E2 and k equations of the form 1 y1 = x1 + (A1,2 x2 + A1,3 x3 + A1,4 x4 + · · · + A1,k xk ), A1,1 1 y2 = x2 + (B2,3 x3 + B2,4 x4 + · · · + B2,k xk ), B2,2 (A.11) 1 y3 = x3 + (C3,4 x4 + · · · + C3,k xk ), C3,3 .. . yk = xk . The xi s can be determined by back-substituting into equation (A.11). A useful feature of Hylleraas’ method is that the sequence of xis provides a practical indication of the convergence of E2 . By changing the number of terms in the trial function it is possible to determine if sufficient accuracy has been obtained or whether the trial function should be extended.


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Appendix B. Calculation of E2 for a confined hydrogen atom

To illustrate the Hylleraas method, the calculation of E2 for a hydrogen atom confined at the centre of an impenetrable sphere of radius 1a0 is presented. A three-term wavefunction ϕ1 = x1 (rc − r) r e−α r + x2 (rc − r) r 2 e−α r + x3 (rc − r) r 3 e−α r

(B.1)

was used for the sake of brevity. The evaluation of the required integrals results in the following matrix elements: ⎛ ⎞ ⎛ ⎞ 0 0.010 905 0.010 225 0.008 349 PB = ⎝0.005 179⎠ , T = ⎝0.010 225 0.011 131 0.010 047⎠ (B.2) 0.007 499 0.008 349 0.010 047 0.009 746 The Hylleraas upper-bound to E2 is given by E2 2 (PB1 x1 + PB2 x2 + PB3 x3 ) + T1,1 x12 +T1,2 x1 x2 + T1,3 x1 x3 + T2,1 x2 x1 + T2,2 x22 + T2,3 x2 x3 + T3,1 x3 x1 + T3,2 x3 x2 + T3,3 x32 , 2 (0x1 + 0.005 179x2 + 0.007 499x3 ) + 0.010 905x12 +0.010 225x1 x2 + 0.008 349x1 x3 2 + 0.010 225x2 x1 + 0.011 131x22 + 0.010 047x2 x3 + 0.008 349x3 x1 2 + 0.010 047x3 x2 + 0.009 746x3 . (B.3) The substitutions 1 T1,2 x2 + T1,3 x3 = y1 − 0.937 643x2 − 0.765 612x3 , x1 = y1 − T1,1 b1 = PB1 = 0, b2 = PB2 = 0.005 179, b3 = PB3 = 0.007 499, T1,2 T1,2 T1,2 T1,3 = 0.001 544, B2,3 = T2,3 − = 0.002 219 = B3,2 , B2,2 = T2,2 − T1,1 T1,1 T1,3 T1,3 B3,3 = T3,3 − = 0.003 354, T1,1

(B.4)

(B.5)

eliminate the xi s, giving E2 2 PB1 y1 + T1,1 y12 + 2 (b2 x2 + b3 x3 ) + B2,2 x2 x2 + B2,3 x2 x3 + B3,2 x3 x2 + B3,3 x3 x3 0.010 905y12 + 2 (0.005 179x2 + 0.007 499x3 ) + 0.001 544x2 x2 + 2 × 0.002 219x2 x3 + 0.003 354x3 x3 .

(B.6)

In a similar fashion, the substitutions x2 = y2 −

B23 x3 = y2 − 1.437 302x3 , B22

c3 = b3 −

B2,3 b2 = 0.000 055, B2,2

C3,3 = B3,3 −

(B.7) B2,3 B2,3 = 0.001 651, B2,2

(B.8)

eliminate the x2 s, leaving E2 2 PB1 y1 + T1,1 y12 + 2b2 y2 + B2,2 y22 + 2c3 x3 + C3,3 x32 .

(B.9)

Differentiating E2 with respect to y1, y2 and x3 gives y1 = −

PB1 = 0, T1,1

y2 = −

b2 = −3.355 149, B2,2

x3 = −

c3 = −0.334 446. C3,3

(B.10)

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E2 can now be written as c2 PB21 b2 − 2 − 3 = 0 − 0.017 376 − 0.000 018 T1,1 B2,2 C3,3 E2 −0.017 394.

E2 −

(B.11)

The wavefunction coefficients follow from equations (B.4) and (B.7) as x2 = y2 − 1.437 302x3 = −2.874 449, x1 = y1 − 0.937 643x2 − 0.765 612x3 = 2.961 264.

(B.12)

References [1] Michels A, de Boer J and Bijl A 1937 Remarks concerning molecular interaction and their influence on the polarisability Physica 4 981–94 [2] Sen K D, Pupyshev V I and Montgomery H E 2009 Exact relations for confined one-electron systems Advances in Quantum Chemistry: Theory of Confined Quantum Systems, Part 1 vol 57 (New York: Academic) pp 25–76 [3] Sternheimer R M 1954 Electronic polarizabilities of ions from the Hartree–Fock wave functions Phys. Rev. 96 951–68 [4] Schiff L I 1955 Quantum Mechanics 2nd edn (New York: McGraw-Hill) pp 151–5 [5] Schiff L I 1995 Quantum Mechanics 2nd edn (New York: McGraw-Hill) pp 171–3 [6] Hylleraas E A 1930 Uber den Grundterm der Zweielektronenprobleme von H−, He, Be++ usw Z. Phys. 65 209—25 Hettema H 2000 Quantum Chemistry (Singapore: World Scientific) pp 124-39 Engl. Transl. [7] Scherr C W and Knight R E 1963 Two-electron atoms: III. A sixth-order perturbation study of the 11S ground state Rev. Mod. Phys. 35 436–42 [8] National Institute of Standards and Technology 2010 Digital Library of Mathematical Functions. Available at http://dlmf.nist.gov/ [9] de Groot S R and ten Seldam C A 1946 On the energy of a model of the hydrogen atom Physica 12 669–82 [10] Aquino N, Campoy G and Montgomery H E 2007 Highly accurate solutions for the confined hydrogen atom Int. J. Quantum Chem. 107 1548–58 [11] Laughlin C 2009 Perturbation theory for a hydrogen-like atom confined within an impenetrable spherical cavity Advances in Quantum Chemistry: Theory of Confined Quantum Systems, Part 1 vol 57 (New York: Academic) pp 203–39 [12] Montgomery H E 2001 Variational perturbation theory of the confined hydrogen atom Int. J. Mol. Sci. 2 103–8 [13] Djajaputra D and Cooper B R 2000 Hydrogen atom in a spherical well: linear approximation Eur. J. Phys. 21 261–7 [14] Ludeña E V 1977 SCF calculations for hydrogen in a spherical box J. Chem. Phys. 66 468–70