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Confined helium on Lagrange meshes Cite this: DOI: 10.1039/c5cp00110b

D. Baye*a and J. Dohet-Eralyb The Lagrange-mesh method has the simplicity of a calculation on a mesh and can have the accuracy of a variational method. It is applied to the study of a confined helium atom. Two types of confinement are considered. Soft confinements by potentials are studied in perimetric coordinates. Hard confinement in impenetrable spherical cavities is studied in a system of rescaled perimetric coordinates varying in [0,1]

Received 9th January 2015, Accepted 23rd February 2015

intervals. Energies and mean values of the distances between electrons and between an electron and

DOI: 10.1039/c5cp00110b

computing times. Pressures acting on the confined atom are also computed. For sphere radii smaller than 1, their relative accuracies are better than 1010. For larger radii up to 10, they progressively

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decrease to 103, still improving the best literature results.

the helium nucleus are calculated. A high accuracy of 11 to 15 significant figures is obtained with small

1 Introduction Many numerical techniques exist for quantum-mechanical calculations in configuration space. Among them, two main qualities may be searched: accuracy and simplicity. However, they are not often encountered simultaneously. For some problems, the Lagrangemesh method has the accuracy of a variational method and the simplicity of a calculation on a mesh.1–5 This approximate variational method involves a basis of Lagrange functions, i.e. infinitely differentiable functions vanishing at all mesh points of a Gauss quadrature, except one. With the help of the associated Gauss quadrature, all matrix elements are very simple. In particular, the matrix elements of the potential are approximated by values of the potential at mesh points like in collocation methods. The striking property of the Lagrange-mesh method is that, in spite of its simplicity, it has essentially the same accuracy as a variational calculation performed with the same Lagrange basis. This performance is not well understood yet.3 However, the accuracy of the method depends on the validity of the Gauss quadrature. Hence, the method can be very bad in the presence of singularities of the potential. This problem can sometimes be cured by a so-called regularization.2–5 The method has been successfully applied to many problems in atomic, molecular and nuclear physics (see ref. 5 for a review). It is particularly useful to solve coupled-channel problems in the continuum6,7 or three-body problems.8,9 The aim of the present paper is to apply the Lagrange-mesh method to a three-body problem: a helium atom confined in some environment.10–16 The confinement can simulate a helium a

Physique Quantique, and Physique Nucle´aire The´orique et Physique Mathe´matique, C.P. 229, Universite´ Libre de Bruxelles (ULB), B-1050 Brussels, Belgium. E-mail: [email protected] b TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada

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gas under pressure or helium atoms trapped in some molecule, carbon cluster or crystal. As we show below, existing techniques17 are very convenient for soft confinements by potentials but can not be applied for an atom confined in an impenetrable spherical cavity. The difficulty arises from simultaneously meeting two different constraints: regularizing the singularities of the three Coulomb terms and forcing the confinement. The former condition is easily treated in perimetric coordinates18,19 which automatically regularize the singularities but which necessarily extend over the whole configuration space. Confinement is easily and accurately treated on a Lagrange mesh over a finite interval for hydrogen.5,20 Here we show that a new coordinate system can be built for helium which keeps the regularization of the potential but over an impenetrable spherical cavity. The principle of the Lagrange-mesh method is recalled in Section 2. In Section 3, the problem of the confined helium atom is presented. In Section 4, the Lagrange-mesh method in perimetric coordinates is recalled. A new system of coordinates is introduced in Section 5 as well as its Lagrange-mesh implementation. Results for the different types of confinement are discussed on Section 6. Concluding remarks are presented in Section 7.

2 Principle of the Lagrange-mesh method Let us consider N mesh points xi associated with a Gaussquadrature approximation,21 ð1 N X FðxÞdx  lk F ðxk Þ: (1) 0

k¼1

The weight coefficients li are also called Christoffel numbers. Lagrange functions are a set of N orthonormal functions fj (x)

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associated with this mesh verifying two conditions.1,4,5 (i) They satisfy the Lagrange property

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fj (xi) = li1/2dij,

(2)

i.e., they vanish at all mesh points, but one. (ii) The Gauss quadrature is exact for products of two Lagrange functions. Let us consider a particle of mass m in a potential V(x). This basis is used in an approximate variational calculation with the trial function cðxÞ ¼

N X

and D1 and D2 are the Laplacians with respect to r1 and r2. For a free atom, the wave functions of the bound states must vanish at infinity. A confinement can be introduced in the problem either by forcing the wave function into some spherical cavity (hard confinement) or by adding a confining potential to H (soft confinement). Hard confinement is obtained with the conditions r1 r R,

(3)

The matrix elements of potential V(x) are calculated at the Gauss approximation as N  X lk fi ðxk ÞV ðxk Þfj ðxk Þ ¼ V ðxi Þdij fi jVj fj 

(4)

k¼1

because of the Lagrange property (2). The variational equations then take the form of mesh equations1,4,5  N  2 X h Tij þ V ðxi Þdij cj ¼ Eci ; (5) 2m j¼1 where the exact or approximate matrix elements Tij = h fi|  d2/ dx2| fji have simple known expressions as a function of the zeros xi and xj. See ref. 1, 4 and 5 for details. This simple approximation can be very accurate with small numbers of mesh points when the Gauss approximation is valid for the potential matrix elements, i.e. when the potential and its derivatives have no singularities. In the presence of singularities, the basis can sometimes be regularized by multiplying the Lagrange functions by a convenient factor R(x),2,4,5 RðxÞ f^j ðxÞ ¼   fj ðxÞ: R xj

(6)

c(R,r2,r12) = c(r1,R,r12) = 0.

3 Confined helium We consider a two-electron atom with an infinite-mass nucleus of charge Ze. This nucleus is fixed and the electrons are characterized by coordinates r1 and r2 with respect to this nucleus. In atomic units h = me = a0 = e = 1, where me is the electron mass and a0 is the Bohr  radius, the Hamiltonian of the helium atom reads 1 1 Z Z 1 H ¼ T þ VC ¼  D1  D2   þ ; 2 2 r1 r2 r12

(10)

Soft confinement can be obtained by adding a potential Vconf (r1,r2) which tends to a large positive constant or to infinity when r1 or r2 tends to infinity. The role of this potential is to reduce the probability density of presence of the electrons at large distances. Let us start with the soft confinement since it can be treated with the same code as for the free atom with only a tiny modification.17

4 Lagrange mesh for soft confinement 4.1

Perimetric coordinates

The system of perimetric coordinates18,19 is very convenient for Lagrange-mesh calculations of three-body systems because the three dimensioned coordinates are independent from each other and vary from zero to infinity. Moreover, the volume element automatically regularizes the singularities of the three Coulomb potentials.17 The perimetric coordinates are composed of three Euler angles and the three coordinates x = r1  r2 + r12, y = r1 + r2 + r12,

The regularized functions ˆfj (x) still verify the Lagrange conditions (2) but they are not orthogonal anymore, in general. However, they are still orthogonal at the Gauss-quadrature approximation and can still be treated as orthonormal in the Lagrange-mesh method without significant loss of accuracy.3,5 Such a regularization is also useful when the particle is confined by an impenetrable wall20 (see Section 5).

(7)

z = r1 + r2  r12.

(11)

The volume element of the dimensioned coordinates reads dV = (x + y)(y + z)(z + x)dx dy dz.

(12)

In perimetric coordinates, the Coulomb potentials become VC ðx; y; zÞ ¼ 

2Z 2Z 2  þ : zþx yþz xþy

(13)

With the volume element dV, the integrand in matrix elements of this potential is bounded everywhere. Hence the Gauss quadrature and the Lagrange-mesh method are accurate. The kinetic-energy operator T is rather complicated.22 It is convenient to write its matrix elements in a symmetric form,17,23 ð1 ð1 ð1 X 3 @F @G hFjTjGi ¼ 2 dx dy dz Aij ðx; y; zÞ ; (14) @x i @xj 0 0 0 i;j¼1 where (x1,x2,x3)  (x,y,z). The coefficients Aij are given by

where r12 = r1  r2

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(9)

The wave function c(r1,r2,r12) of an S state must thus verify cj fj ðxÞ:

j¼1



r2 r R.

(8)

A11 = x( y + z)(x + y + z) + xz(z + x),

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i.e., they vanish at all points of the three-dimensional mesh, but one. With the volume element (12), they are not orthogonal but the scalar product hFi 0 j 0 k 0 |Fijki is calculated with the Gaussquadrature approximation as dii 0 djj 0 dkk 0 . They are thus treated as an orthonormal basis in the method. The kinetic-energy matrix elements are given by

A22 = yz( y + z) + y(z + x)(x + y + z), A33 = yz( y + z) + xz(z + x), A12 = A21 = 0,

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A13 = A31 = xz(z + x), A23 = A32 = yz( y + z).

4.2

(15)

Lagrange mesh and functions

n

The Lagrange–Laguerre functions are defined as1 fj ðxÞ ¼ ð1Þ j xj 1=2

LN ðxÞ x=2 e ; x  xj

þ dii0 dkk0

LN(xi) = 0.

þ dii0 djj0

(17)

Notice that the Lagrange functions are linearly independent polynomials of degree N  1 multiplied by an exponential which is the square root of the Laguerre weight function exp(x). The basis is thus equivalent, for example, to a basis formed of the Laguerre polynomials of degrees 0 to N  1 multiplied by exp(x/2). The functions fj (x) are associated with the Gauss–Laguerre quadrature.21 They verify the Lagrange conditions (2) and integrals of products of two Lagrange functions are exactly given by the Gauss quadrature since the integrand is the product of the Laguerre weight function exp(x) by a polynomial of degree 2N  2.24 Functions (16) are exactly orthonormal over (0,N). The first derivative of a one-dimensional Lagrange-Laguerre function at mesh points is given by rffiffiffiffi xj 1 0 li 1=2 fj ðxi Þ ¼ ð1Þij (18) xi xi  xj for i a j and by 0

1 : 2xi

(19)

X

  ðNy Þ 0 ðN y Þ 0 A22 hx xi ; hy yn ; hz zk mn fj ðyn Þfj 0 ðyn Þh2 y

n

(16)

where LN(x) is the Laguerre polynomial of degree N and the xi are its zeros,

li 1=2 fi ðxi Þ ¼ 

  1=2 1=2 Fi0 j 0 k0 jTjFijk  2N i0 j 0 k0 N ijk hx hy hz ( X   ðN Þ 0 ðN Þ 0  djj 0 dkk0 A11 hx xn ; hy yj ; hz zk ln fi x ðxn Þfi0 x ðxn Þh2 x

X

ðNz Þ 0

A33 ðhx xi ; hy yj ; hz zn Þn n fk

ðNz Þ 0

ðzn Þfk0

ðzn Þh2 z

n

  1=2 ðNx Þ 0 ðNy Þ 0   þ dkk0 A12 hx xi ; hy yj 0 ; hz zk li mj0 fi 0 ðxi Þfj yj 0    1=2 ðNx Þ 0 1 ðNy Þ 0    hx hy fi ðxi0 Þfj0 yj þ A12 hx xi0 ; hy yj ; hz zk li0 mj h   ðN Þ 0 ðN Þ 0 þ djj 0 A13 hx xi ; hy yj ; hz zk0 ðli n k0 Þ1=2 fi0 x ðxi Þfk z ðzk0 Þ i   ðN Þ 0 ðN Þ 0 þ A13 hx xi0 ; hy yj ; hz zk ðli0 n k Þ1=2 fi x ðxi0 Þfk0 z ðzk Þ ðhx hz Þ1

  1=2 ðNy Þ 0   ðNz Þ 0 þ dii0 A23 hx xi ; hy yj ; hz zk0 mj n k0 fj 0 yj fk ðzk0 Þ    1=2 ðNy Þ 0   ðNz Þ 0  1 o fj yj0 fk0 ðzk Þ hy hz : þ A23 hx xi ; hy yj0 ; hz zk mj0 n k (23) From now on, we consider Nx = Ny = N which implies xi  yi, li  mi, and hx = hy = h. The Lagrange functions are used as a variational basis to expand an S-wave trial function, cðx; y; zÞ ¼

Nz N X is X X

  1=2 Cijk 2 1 þ dij (24)

i¼1 j¼1 k¼1

Three-dimensional mesh and basis are obtained as follows. Let xp ( p = 1,. . .,Nx), yq (q = 1,. . .,Ny) and zr (r = 1,. . .,Nz) be the zeros of Laguerre polynomials with respective degrees Nx, Ny and Nz. Three-dimensional Lagrange functions Fijk(x,y,z) associated with the mesh (hxxp,hy yq,hzzr) are defined by   Fijk ðx; y; zÞ ¼ N ijk 1=2 fi ðNx Þ ðx=hx Þfj ðNy Þ y hy fk ðNz Þ ðz=hz Þ: (20) The functions f (N) i are given by expression (16) with N replaced by Nx, Ny or Nz. The corresponding Christoffel numbers are denoted as li, mj and nk. Scale parameters hx, hy and hz are introduced in order to fit the different meshes to the size of the actual physical problem. The normalization factor Nijk is defined as Nijk = hxhyhz(hxxi + hy yj)(hxxi + hzzk)(hy yj + hzzk).

(21)

The Lagrange functions Fijk(x,y,z) satisfy the Lagrange property Fijk(hxxp,hy yq,hzzr) = (Nijklimjnk)1/2dipdjqdkr,

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(22)



 Fijk ðx; y; zÞ  Fjik ðx; y; zÞ ; where s = 0 in the symmetric case and 1 in the antisymmetric case, and j r i  s because of the symmetry with respect to the exchange of electrons 1 and 2. The matrix representing the total potential V = VC + Vconf is immediately obtained with a triple Gauss quadrature and the Lagrange property (22) as hFi 0 j 0 k 0 |V|Fijki E V(hxi,hxj,hzzk)dii 0 djj 0 dkk 0

(25)

The variational calculation then reduces to mesh-like equations Nz n N X is X X

1þdij

1=2  1=2     1þdi0 j 0 Fi0 j 0 k0 jTjFijk  Fi0 j 0 k0 jTjFjik

i¼1 j¼1 k¼1

 

 þ V hxi ;hxj ;hz zk E dii0 djj0 dkk0 Cijk ¼0: (26)

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Energies and wave functions are obtained from the eigenvalues and eigenvectors of a large sparse symmetric matrix. Mean values of a multiplicative operator O(x,y,z) are simply given at the Gauss approximation by

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hcjOðx; y; zÞjci 

Nz N X is X   1X 2 Cijk O hxi ; hxj ; hz zk 2 i¼1 j¼1 k¼1



(27)



þO hxj ; hxi ; hz zk :

The integrands in its matrix elements are automatically regularized by the volume element. The kinetic matrix elements become ð1 ð1 ð1 X 3 @F @G hFjTjGi ¼ 2ð2RÞ4 du dv dw Bij ðu; v; wÞ (35) @u i @uj 0 0 0 i;j¼1 where

5 Lagrange mesh for hard confinement 5.1

The Coulomb potential reads

 1 Z Z 1 VC ðu; v; wÞ ¼   þ : R u þ w  uw v þ w  vw ðu þ vÞð1  wÞ (34)

B11 = u(1  w)[avb + (1  u)2wau],

Rescaled perimetric coordinates

B22 = v(1  w)[(1  v)2wav + aub],

In an impenetrable spherical cavity of radius R, the perimetric coordinates are constrained by

B33 = w(1  w)3[vav + uau],

0 r x r 2R  z,

B12 = B21 = uvw(1  w)[(v  1)av + (u  1)au],

0 r y r 2R  z, 0 r z r 2R.

B13 = B31 = uw(1  w)2[vav + (u  1)au], (28)

Hence the kinetic matrix elements (14) must be rewritten as

B23 = B32 = vw(1  w)2[(v  1)av + uau] with

ð 2R ð 2Rz ð 2Rz X 3 @F @G hFjTjGi ¼ 2 dz dx dy Aij ðx; y; zÞ : (29) @x i @xj 0 0 0 i;j¼1 The z dependence of the upper bounds of the x and y coordinates is a difficulty for the use of the Lagrange-mesh method. We thus introduce a new set of coordinates (u,v,w)  (u1,u2,u3) defined over [0,1] by x ; 2R  z y ; v¼ 2R  z z w¼ : 2R

u ¼

(30)

au = u + w  uw, av = v + w  vw, b = u + v + w  uw  vw + uvw. 5.2

Lagrange mesh and functions

Let us introduce a convenient basis over the [0,1] interval. Regularized Lagrange–Legendre functions are defined by rffiffiffiffiffiffiffiffiffiffiffiffi uj PN ð2u  1Þ ð1  uÞ; (38) fj ðuÞ ¼ ð1ÞNj 1  uj u  uj

PN(2ui  1) = 0. (r1 = R),

v = 1 (r2 = R), w=1

(r1 = r2 = R, r12 = 0).

(31)

Inversely, one has x = 2Ru(1  w), y = 2Rv(1  w), z = 2Rw.

(32)

The volume element then becomes dV = (2R)6(u + v)(u + w  uw)(v + w  vw)(1  w)3du dv dw. (33)

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(37)

where the mesh points ui are the zeros of the shifted Legendre polynomial of degree N,

Their upper values are indeed u=1

(36)

(39)

They are associated with the Gauss–Legendre quadrature on the [0,1] interval. The regularized Lagrange functions (38) correspond to standard Lagrange–Legendre functions,1 shifted25 and multiplied by the factor (1  u)/(1  uj), so that they vanish at u = 1. In ref. 25 on the contrary, they are multiplied by u/uj so that they vanish at u = 0. In ref. 20, they are multiplied by both factors simultaneously and vanish at u = 0 and 1. Notice that all these types of regularized functions satisfy the Lagrange conditions (2) but are not orthogonal. Nevertheless they can be treated as orthonormal in Lagrangemesh calculations without significant loss of accuracy.3,5 The first derivative of a Lagrange-Legendre function (38) at mesh points is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uj ð1  ui Þ 1 iþj 1=2 0   (40) li fj ðui Þ ¼ ð1Þ ui 1  uj ui  uj

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for i a j and by 0

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li 1=2 fj ðui Þ ¼ 

1 : 2ui ð1  ui Þ

(41)

Over a three-dimensional mesh (up,vq,wr), where up, vq, wr are solutions of eqn (39) with possibly different values Nu, Nv, Nw of N, three-dimensional Lagrange functions Fijk(u,v,w) are defined by 1=2 ðNu Þ

Fijk ðu; v; wÞ ¼ N ijk fi

ðNv Þ

ðuÞfi

ðNw Þ

ðvÞfi

ðwÞ

  Fi0 j 0 k0 jVC jFijk  dii0 djj0 dkk0 " # 1 Z Z 1     þ  : R ui þ wk  ui wk uj þ wk  uj wk ui þ uj ð1  wk Þ (48) The Lagrange-mesh equations are

(42)

Nw n N X is X X 

with

1 þ dij

1=2  1=2 1 þ di0 j 0

i¼1 j¼1 k¼1 6

    Fi0 j0 k0 jTjFijk  Fi0 j0 k0 jTjFjik  

 þ VC ui ; uj ; wk  E dii0 djj 0 dkk0 Cijk ¼ 0:

3

Nijk = (2R) (ui + vj)(ui + wk  uiwk)(vj + wk  vjwk)(1  wk) . (43)



These functions satisfy the Lagrange conditions Fijk(up,vq,wr) = (Nijklimjnk)1/2dipdjqdkr,

(44)

where li, mj and nk are now the weights of the Gauss–Legendre quadrature over the [0,1] interval with Nu, Nv and Nw points, respectively. They are not orthogonal but they are orthonormal at the Gauss-quadrature approximation and are treated as an orthonormal basis in the method, hFi 0 j 0 k 0 |Fijki - dii 0 djj 0 dkk 0 .

ðNv Þ 0

B22 ðui ; vn ; wk Þmn fj

ðNv Þ 0

ðvn Þfj0

þ dii0 djj0





ðNw

B33 ui ; vj ; wn n n fk

Þ0

ðNw

ðwn Þfk0

Þ0

6.1

ðwn Þ

n

(46)

i   1=2 ðNu Þ 0 ðN Þ 0   ðui0 Þfj 0 u vj þ B12 ui0 ; vj ; wk li0 mj fi h   ðN Þ 0 ðN Þ 0 þ djj 0 B13 ui ; vj ; wk0 ðli n k0 Þ1=2 fi0 u ðui Þfk w ðwk0 Þ i   ðN Þ 0 ðN Þ 0 þ B13 ui0 ; vj ; wk ðli0 n k Þ1=2 fi u ðui0 Þfk0 w ðwk Þ h   1=2 ðNu Þ 0   ðNw Þ 0 fj 0 vj fk ðwk0 Þ þ dii0 B23 ui ; vj ; wk0 mj n k0 io   1=2 ðNu Þ 0   ðNw Þ 0 fj vj0 fk0 ðwk Þ : þ B23 ui ; vj 0 ; wk mj 0 n k From now on, we take Nu = Nv = N. An S-wave trial function is expanded as cðu; v; wÞ ¼

Nw N X is X X

  1=2 Cijk 2 1 þ dij

i¼1 j¼1 k¼1

(47)



 Fijk ðu; v; wÞ  Fjik ðu; v; wÞ ; where s is defined as above. The potential matrix elements simply read

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(50)



All results are presented in atomic units except in Table 7. An error of a few units may affect the last displayed digit.

ðv n Þ

h   1=2 ðNu Þ 0 ðN Þ 0   þ dkk0 B12 ui ; vj 0 ; wk li mj 0 fi 0 ðui Þfj u vj 0

Nw N X is X   1X C2 O ui ; uj ; wk 2 i¼1 j¼1 k¼1 ijk

6 Results

n

X

hcjOðu; v; wÞjci 

þ O uj ; ui ; wk :

n

X

Notice that obtaining energies does not require calculating eigenvalues of a generalized eigenvalue problem since the basis is treated as orthonormal. Mean values of a multiplicative operator O(u,v,w) are given at the Gauss approximation by

(45)

The kinetic matrix elements are given by   1=2 1=2 Fi0 j 0 k0 jTjFijk  2ð2RÞ4 N i0 j0 k0 N ijk ( X   ðN Þ 0 ðN Þ 0  djj0 dkk0 B11 un ; vj ; wk ln fi u ðun Þfi0 u ðun Þ þ dii0 dkk0

(49)

Soft confinement

The study of soft confinement is performed in perimetric coordinates using the code of ref. 17 with two modifications. (i) A confining potential Vconf is added to the Coulomb potential; this is a very simple addition in the code since no matrix elements are needed. (ii) The search of the lowest eigenvalues of the rather scarce Hamiltonian matrix is now performed with the Jacobi–Davidson algorithm.26 Mean values are calculated with eqn (27) for the distance r12 between electrons and for the distances r1 and r2 between the electrons and the nucleus. The calculations are performed to provide 12–15 significant figures in the fastest way, when such an accuracy can be reached with reasonable basis sizes. The calculation starts with a search for an optimal domain of the scale parameters h and hz. The significant digits are obtained by comparison between several calculations with different numbers of mesh points. The simplest of the calculations giving the requested accuracy is then kept. As a first penetrable confinement, we choose like in ref. 12 the harmonic potential  1  Vconf ðr1 ; r2 Þ ¼ o2 r1 2 þ r2 2 ; 2

(51)

where o is a parameter. The convergence for o = 1 is studied in Table 1 as a function of the numbers N and Nz. The total mesh 1 size NT ¼ NðN þ 1ÞNz is also given. 2

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Table 1 Convergence of the ground-state energy and the mean interparticle distances of a helium atom confined by the harmonic potential (51) with o = 1 as a function of the numbers N and Nz of mesh points and the total mesh size NT

N

Nz

NT

10

10

550

15

15

1800

20

20

4200

25

20

6500

25

25

8125

30

25

11 625

30

30

13 950

h

hz

E

hr12i

hr1i = hr2i

0.15 0.20 0.15 0.20 0.15 0.20 0.15 0.20 0.15 0.20 0.15 0.20 0.15 0.20

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

2.0730759 2.0730353876 2.0730353620325 2.0730353620477 2.07303536205189 2.07303536205154 2.07303536205195 2.07303536205188 2.07303536205194 2.07303536205186 2.07303536205194 2.07303536205194 2.07303536205193 2.07303536205191 2.073035

1.0858179 1.08568626707 1.08568576880716 1.08568576860892 1.08568576862421 1.08568576862456 1.08568576862425 1.08568576862425 1.08568576862419 1.08568576862399 1.08568576862413 1.08568576862422 1.08568576862421 1.08568576862423

0.7237152 0.7236444069 0.72364414179632 0.72364414169294 0.72364414170093 0.72364414170114 0.72364414170096 0.72364414170096 0.72364414170092 0.72364414170080 0.72364414170088 0.72364414170093 0.72364414170091 0.72364414170093

Ref. 12

The parameters h and hz are roughly optimized. One observes a fast convergence for h = 0.15–0.20 and hz = 0.15–0.20 where very accurate results are already obtained for N = Nz = 20. The results are quite insensitive to variations of hz in this interval. Rounding errors start to slightly deteriorate the accuracy around N = Nz = 30. For small values of o, the convergence is very similar to the free-atom case. See e.g. Table 1 of ref. 17. The Lagrange-mesh results for various o values are presented in Table 2. For each case, the computing time of the lowest eigenvalue is smaller than one minute on a fast PC for the largest numbers of mesh points used. The computing time of the mean values is negligible. The values for the free atom (o = 0) have an accuracy of about 1013 like in ref. 17. A relative accuracy of 1012 is already obtained with N = Nz = 20. The confinement reduces the size of the atom when o increases while increasing its energy. The scale factors for the weakest confinement are inspired by those of a free atom. When o increases, they progressively decrease. To obtain a constant accuracy (thirteen significant digits), the numbers N and Nz can both decrease when the confinement becomes stronger. The present energies confirm the six-digit results of ref. 12 for o r 1, up to a possible rounding. As a second potential with a penetrable confinement, we choose like in ref. 12 2

2

2

2

Vconf (r1,r2) = V0(2  er1/R  er2/R ),

The convergence is studied in Table 3 for V0 = 100 and R = 4 as a function of N and Nz. Good values of the scale factors are much smaller than for the free atom.17 Here they are roughly optimized as h = 0.1, hz = 0.15. The convergence is very fast. Good results are already obtained with N = Nz = 10. An accuracy of about 14 digits is obtained with N = Nz = 20 for the energy and for the mean distances. The Lagrange-mesh energies and mean distances are displayed in Table 4 for V0 = 25 and 100 and some values of R. Here also, the computing times are around one minute or less. The scale parameters are rather small for R = 1 and must increase with R. The ground-state energy is positive for the smaller R values but the three-body system is nevertheless deeply bound with respect to the asymptotic value 2V0 of the confinement potential. Hence the wave functions decrease rather fast. Even for R = 10, the properties significantly differ from the free atom. The mean distances are scaled down with respect to the free atom17 but hr12i and hr1i indicate that the confinement does not change much the shape of the electron distribution. Indeed, the ratio hr12i/hr1i progressively increases from the value 1.45 at strong confinement (R = 1, V0 = 100) to the free value 1.53. Our results confirm the four- and six-digit energies of ref. 12 except for their last two digits displayed at R = 4 and 5. 6.2

(52)

Hard confinement

Now we consider a nucleus confined in an impenetrable sphere. There are no scale parameters as the size is fixed by

where V0 and R are parameters.

Table 2 Ground-state energy and mean interparticle distances of a helium atom confined by the harmonic potential (51) as a function of the numbers N and Nz of mesh points and the scale parameters h and hz

o

N

Nz

h

hz

E

hr12i

hr1i = hr2i

0 0.01 0.05 0.1 0.25 0.5 1 2 5 10

30 30 30 25 25 25 25 20 20 20

25 25 25 25 25 20 20 20 20 20

0.30 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.10 0.06

0.35 0.35 0.35 0.30 0.25 0.20 0.20 0.15 0.10 0.06

2.9037243770340 2.9036050414229 2.9007485071345 2.8919107031034 2.8330693153970 2.6487031494197 2.0730353620519 0.4931738615042 5.5550214188965 17.1621913740574

1.4220702555659 1.4219400165127 1.4188596636545 1.4097410240940 1.3596818036283 1.2563677352972 1.0856857686242 0.8792960764484 0.6193775846175 0.4594214282815

0.9294722948737 0.9293956002920 0.9275805434260 0.9221955320482 0.8923612390213 0.8296645227059 0.7236441417010 0.5923199939980 0.4228045127786 0.3162727498955

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Table 3 Convergence of the ground-state energy and the mean interparticle distances of a helium atom confined by potential (5.2) with V0 = 100 and R = 4. The scaling parameters are h = 0.10 and hz = 0.15

N

Nz

NT

E

hr12i

hr1i = hr2i

10 15 20 25 Ref. 12

10 15 20 25

950 1800 4200 8125

2.3895176 2.389521083273 2.38952108337013 2.38952108337011 2.389531

0.7172396 0.717240933789 0.71724093382109 0.71724093382111

0.4870826 0.487083296959 0.48708329696605 0.48708329696608

Table 4 Ground-state energy and mean interparticle distances of a helium atom confined by potential (52) as a function of the numbers N and Nz of mesh points and the scale parameters h and hz

R

N

Nz

h

hz

E

hr12i

hr1i = hr2i

V0 = 25 1 2 3 4 5 10 25 100

20 20 20 20 20 20 25 25

20 20 20 20 20 20 20 20

0.10 0.10 0.10 0.10 0.15 0.20 0.30 0.30

0.10 0.10 0.15 0.15 0.15 0.30 0.35 0.35

9.0056649196150 2.2056338218026 0.0586101621097 0.9379748742074 1.4934630265876 2.4403181864333 2.8145748135893 2.8977889060887

0.5740154098032 0.7325744342996 0.8431468640408 0.9261396704923 0.9912659220321 1.1805989951242 1.3466988063927 1.4157415359464

0.3921371725028 0.4968874426881 0.5688300180991 0.6222657138458 0.6638539995004 0.7828965128863 0.8845542404093 0.9257410632589

V0 = 100 1 2 3 4 5 10 25 100

25 20 20 20 20 25 25 25

25 20 20 20 20 20 20 20

0.08 0.10 0.10 0.10 0.10 0.20 0.30 0.30

0.08 0.10 0.10 0.15 0.15 0.25 0.35 0.35

25.9833284621647 9.9193413237201 4.8125898345331 2.3895210833701 1.0066877980279 1.4748724235163 2.5868427061834 2.8803231455112

0.4079160085829 0.5436276792868 0.6408754493044 0.7172409338211 0.7800447623969 0.9859082288883 1.2308212121751 1.3985909114282

0.2813631502792 0.3724398341115 0.4369052486715 0.4870832969661 0.5280608359132 0.6605186296182 0.8139656785849 0.9155880710520

Table 5 Convergence of the ground-state energy and the mean interparticle distances of a helium atom confined in a sphere of radius R as a function of the numbers N and Nw of mesh points and the total mesh size NT

N

Nw

R = 0.1 8 10 10 15 15 Ref. 14

8 10 15 15 20

288 550 825 1800 2400

10 15 15 20 20 25 30 35

550 825 1800 2400 4200 5250 9750 16 275

15 20 25 30 30 35 35

1800 2400 5250 9750 13 950 16 275 22 050

R=1 10 10 15 15 20 20 25 30 Ref. 10 Ref. 12 and 14 R = 10 15 15 20 25 30 30 35 Ref. 12 and 16

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NT

E

hr12i

hr1i = hr2i

906.5624079 906.562422907 906.562422919887 906.562429919886 906.562422919888 906.562423

0.0695803858 0.06958038288807 0.06958038288417 0.06958038288417 0.06958038288416

0.0495012410 0.0495012463326 0.049501246340126 0.049501246340120 0.049501246340121

0.64366425393 0.6436642538710 0.6436642538780 0.6436642538779 0.6436642538780 0.6436642538778 0.6436642538788 0.6436642538769 0.643938

0.441796632210 0.44179663209855 0.44179663210328 0.44179663210331 0.44179663210334 0.44179663210341 0.44179663210334 0.44179663210324

1.42076796281 1.42207279004 1.42207019674 1.42207017366 1.42207017298 1.42207017234 1.42207017296

0.92864575591 0.92947432257 0.92947226315 0.92947225158 0.92947225122 0.92947225087 0.92947225121

1.01575497553 1.01575497590 1.01575497604833 1.01575497604841 1.01575497604864 1.01575497604866 1.01575497604867 1.01575497604854 1.015870 1.015755 2.90479 2.9037272 2.9037243826 2.903724375625 2.903724375691 2.903724375668 2.903724375687 2.903724

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Table 6 Ground-state energy and mean interparticle distances of a helium atom confined in a sphere of radius R as a function of the numbers N and Nw of mesh points

R

N

Nw E

hr12i

hr1i = hr2i

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

15 15 15 15 15 15 15 15 15 15 20 25 25 30 30 35 35 35 35

15 906.562422919888 20 206.151712932762 20 82.334517028118 20 40.9802803317051 20 22.7413028191335 20 13.3181272418281 20 7.9252160465796 20 4.6104075542394 20 2.4632359886210 20 1.0157549760484 20 2.6040382751762 25 2.8724948864757 25 2.9004857633632 30 2.9034108492031 30 2.9036959089307 35 2.9037219115318 35 2.9037241707136 35 2.903724360207 35 2.903724375687

0.0695803828842 0.1383657890250 0.206218777530 0.273001759106 0.338577477653 0.402809660984 0.465563856457 0.526708460703 0.586115943465 0.643664253878 1.097202490172 1.322925949994 1.399878118648 1.418255199620 1.421528373775 1.422002693115 1.422062567876 1.422069437714 1.422070172936

0.0495012463401 0.0979699845849 0.1453521119299 0.1915916590472 0.2366312132559 0.2804124554645 0.3228768193965 0.3639662772674 0.4036242499908 0.4417966321033 0.7339563805892 0.871985220059 0.916962790629 0.927363648349 0.929176980774 0.929435862703 0.929468183475 0.929471860371 0.929472251212

the radius R. Otherwise, the code for the new coordinate system follows the same philosophy as the previous one. The total 1 mesh size is NT ¼ NðN þ 1ÞNw for singlet states and NT ¼ 2 1 NðN  1ÞNw for triplet states. The mean values are calculated 2 from (50) with (32). The convergence of Lagrange-mesh results for the ground state is studied in Table 5 for various values of R, as a function of N and Nw. For large R, the convergence is rather slow but it is extremely fast for small R. One obtains 15 significant figures for R = 0.1 and 1 but only 11 for R = 10. The situation is reversed with respect to soft-confinement calculations in perimetric coordinates where larger bases are needed at small R. The results are compared with ref. 10, 12, 14 and 16. A good agreement with six significant digits is observed for the most recent results.

Table 7

Energies and mean distances for the ground state are gathered in Table 6 for a number of R values. The CPU times increase with R from a few seconds to about 500 seconds at N = Nw = 35. The energies confirm the six-digit variational energies of ref. 12 from 0.5 to 10 and ref. 16 from 1 to 10. The present coordinate system allows a better description of smaller R values. The earlier variational energies of ref. 10 have an absolute accuracy below 0.001. This reference is the only one to present hr12i mean distances. Their accuracy increases from 104 to 0.002 when R varies from 0.5 to 6. From the radius dependence of the energies E(R), one can deduce the pressure acting on the confined atom,10,12,16 P¼

1 dE : 4pR2 dR

(53)

The derivative is performed numerically with a 4-point finitedifference formula. The results are presented in Table 7. As before, the significant digits are estimated by comparison between several calculations differing by the number of mesh points. Only those digits are displayed in Table 7. The results are presented both in atomic units and in atmospheres (1 a.u. = 2.903628236775  108 atm). A high accuracy is reached up to R = 2. Beyond that value, the differences between neighboring energies become tiny and the relative error increases. The results are compared with several earlier variational calculations. The pressures of ref. 10 have a relative accuracy better than 104 at R = 0.5 and still better than 103 up to R = 2. They become quite poor beyond R = 4 as could be expected from the comparison between their variational calculations with different basis sizes. The results of ref. 12 are consistent with ours for the three displayed digits except at R = 5 and 8. The pressures of ref. 16 have a relative accuracy varying between 0.001 and 0.006. Energies and mean distances for the first excited singlet level are presented in Table 8. The energies are given by the second lowest eigenvalue of the Hamiltonian matrix of eqn (49), like in variational calculations.27,28 The calculation time of the

Pressure acting on a helium atom confined in a sphere of radius R in a.u. and atm. The powers of ten are indicated between brackets

R

N

Nw

P (a.u.)

P (atm)

Ref. 10

Ref. 12

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

20 20 20 20 20 20 20 20 20 20 25 25 25 30 30 35 35 35 35

20 20 20 20 20 20 20 20 20 20 25 25 25 30 30 35 35 35 35

1.50742673864[5] 4.51288006460[3] 5.6829402825[2] 1.2871182851[2] 4.0172156095[1] 1.5344287331[1] 6.7323163746 3.2668806106 1.7109177385 9.5100856621[1] 1.383499912[2] 6.19283139[4] 3.70224159[5] 2.36511117[6] 1.5262285[7] 9.871147[9] 6.39998[10] 4.1614[11] 2.711[12]

4.37700684319[13] 1.31037259848[12] 1.6501145872[11] 3.7373129968[10] 1.1664500677[10] 4.4554105969[9] 1.9548143924[9] 9.4858067871[8] 4.9678690563[8] 2.7613753263[8] 4.017169411[6] 1.79816801[5] 1.07499332[4] 6.86740358[2] 4.4316001[1] 2.866214 1.85831[1] 1.2083[2] 7.874[4]

1.16640[10] 4.45436[9] 1.95450[9] 9.48410[8] 4.96703[8] 2.76049[8] 4.01456[6] 1.79150[5] 1.08677[4] 9.02194[2] 1.33015[2]

1.167[10] 4.455[9] 1.954[9] 9.485[8] 4.967[8] 2.762[8] 4.018[6] 1.798[5] 1.074[4] 6.882[2] 4.434[2]

Phys. Chem. Chem. Phys.

Ref. 16

4.00729[6] 1.79391[5] 6.8503[2] 2.84795

1.868[1]

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Table 8 Energy and mean interparticle distances of the 21S singlet level of a helium atom confined in a sphere of radius R

R

N Nw E

hr12i

hr1i = hr2i

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

15 15 15 15 15 15 15 15 15 15 20 25 25 30 30 30 30 35 35

0.0774489892860 0.1550472684474 0.2329635031645 0.311436535890 0.390828529869 0.471731678057 0.555200036019 0.643293324656 0.740350389720 0.853808811249 1.343903122341 1.866351750214 2.408986509893 2.94863042083 3.43587896714 3.85819002636 4.21584257324 4.51057216717 4.74471244944

0.05900499443503 0.11767957674569 0.17601138786641 0.23398345140640 0.2915687438667 0.3487162262547 0.4053102256263 0.4610317869316 0.5147758124513 0.561631072086 0.883824224122 1.203075085099 1.498595969032 1.782952066008 2.036478158792 2.254322411040 2.437665074483 2.588081892182 2.707196655205

15 1963.75792204146 20 477.023366387052 20 205.784321398280 20 112.226705249011 20 69.550421013503 20 46.7055303405089 20 33.1313908859934 20 24.4478624291552 20 18.5745828808285 20 14.4137660915523 25 0.9465884738869 25 1.1141215137421 25 1.7175175902269 30 1.9497613214615 30 2.0507021791853 30 2.0980850744321 30 2.1215112224588 35 2.1334532924803 35 2.1396198861786

lowest two eigenvalues is close to twice the calculation time of the ground-state energy alone. Obtaining the same accuracy as for the ground state sometimes requires higher numbers of mesh points. At R = 0.1, the excitation energy exceeds 1000 a.u. For R = 3, 5, 7 and 10, a comparison is possible with ref. 16. The six-digits results of ref. 16 agree with the present ones at the level of 106. The best variational energies of ref. 13 have an accuracy around one percent. Energies and mean distances for the lowest triplet level are gathered in Table 9. They are obtained from the lowest-energy solution for the spatially antisymmetric state. CPU times reach two minutes at R = 10. The numbers of mesh points are very close to those for the ground state. At R = 0.1, the excitation energy reaches about 1500 a.u. The 23S level is thus above the 21S level contrary to the free-atom situation as already observed in ref. 13. This is probably due to the additional constraint that

Table 9

the wave function must vanish for r1 = r2 in a tiny space. The usual ordering is recovered around R = 1 with an excitation energy of about 5 a.u. At all confinements, the triplet state is slightly less extended than the singlet excited state like for the free atom.29 The best variational energies of ref. 13 reach an accuracy of about 104 above R = 1. Below that value the perturbation approach is more accurate; its relative accuracy improves from about 103 at R = 1 to about 104 at R = 0.1.

7 Concluding remarks Various types of helium confinement can be accurately treated with the Lagrange-mesh method. They are based on the perimetric coordinates for soft confinements or on a new coordinate system for the hard confinement. The method rapidly leads to a large but sparse symmetric matrix. The eigenvalue problem is not generalized, even with the new coordinate system for which the basis is not orthogonal. This striking property results from the systematic use of the Gauss quadrature associated with the Lagrange basis. This simplifying approximation does not cost accuracy.3,5 The present results improve previous works by at least five orders of magnitude for the energies and allow a very accurate calculation of mean interparticle distances. This is obtained with short computer times on a personal computer. The scale parameters given in the present tables allow avoiding a search for their optimal domain of values in future similar calculations. Together with the energies, approximate wave functions are also available. Their high accuracy is confirmed by the high accuracy on the mean values of the coordinates. They are available for other applications. A serious challenge is the extension of the method to more than three charged particles because a convenient coordinate system, i.e. where the different Coulomb singularities can be regularized, is not available yet.

Energy and mean interparticle distances of the 23S triplet level of a helium atom confined in a sphere of radius R

R

N

Nw

E

hr12i

hr1i = hr2i

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

15 15 15 15 15 15 15 15 15 15 15 20 20 20 30 30 30 30 30

15 15 15 15 15 15 15 15 15 15 20 25 30 30 30 30 30 30 30

2370.72702330397 568.187610564046 241.491539024040 129.543621160698 78.8208637033835 51.8567199330814 35.9511073837302 25.8555040261363 19.0892330327807 14.3597149208699 0.56025123373470 1.37051061013706 1.87461159637791 2.04804409351074 2.11781628516207 2.14856447005490 2.16278382665888 2.16948105237618 2.17262725224936

0.07423593580689 0.14810698419629 0.22158029618099 0.29462361098139 0.3672053489194 0.4392947116545 0.5108617890944 0.5818776720090 0.6523145690865 0.7221459268468 1.3824276676122 1.965771607822 2.474178173270 2.91656252964 3.29808901533 3.61901294888 3.87826184464 4.07681612079 4.21955450184

0.04974687348320 0.09897862549715 0.14768195605843 0.19584362548901 0.2434505310533 0.2904897901855 0.3369488298349 0.3828154812295 0.4280780791261 0.4727255643597 0.8835721456814 1.2285886749956 1.515265802863 1.755588974154 1.957801932623 2.125320088136 2.259362209113 2.36138847941 2.43442417024

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Acknowledgements This text presents research results of the interuniversity attraction pole programme P7/12 initiated by the Belgian-state Federal Services for Scientific, Technical and Cultural Affairs. TRIUMF receives funding via a contribution through the National Research Council Canada.

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