Helium atom under pressure

Hard-sphere confinement is used to study helium atoms under pressure. The confined-helium Schrodinger equation is solved with a high accuracy by a Lagrange-mesh method.

The effects of high pressure on a helium gas can be estimated by studying the helium atom in a hard confinement, i.e. confined at the centre of an impenetrable spherical cavity, for different cavity radii.Contrary to the free helium atom, the confined helium has not been described with a very high accuracy until recently, when we have developed a Lagrange-mesh method to study this system [1].This method improves by several order of magnitudes the accuracy of previous approaches [2][3][4].
The outlines of the model are the following.The assumed-infinite-mass nucleus is fixed and the electrons are characterized by coordinates r 1 and r 2 with respect to this nucleus.In atomic units, the Hamiltonian of the helium atom reads where r 12 = r 1 − r 2 and Δ 1 and Δ 2 are the Laplacians with respect to r 1 and r 2 .The confinement is introduced by forcing the wave function into some spherical cavity of radius R (r 1 , r 2 ≤ R).The wave function ψ(r 1 , r 2 , r 12 ) of an S state must thus verify the Schrödinger equation and vanishes at r 1 = R and r 2 = R.The coordinates (r 1 , r 2 , r 12 ) are advantageously replaced by the coordinates (u, v, w) defined over [0,1] by The confinement implies that the wave function ψ(u, v, w) vanishes at u = 1, v = 1, and w = 1.The Schrödinger equation is solved by the Lagrange-mesh method [5][6][7], an approximate variational approach taking the form of a system of mesh equations by computing the Hamiltonian and overlap matrix elements with a Gauss quadrature.Using the coordinates (u, v, w) is essential for an easy treatment of the confinement and an high accuracy of the Gauss quadrature.The ground-state energy and wave function are obtained by diagonalizing a rather large (matrix dimension≈ 10 3 ∼ 2×10 4 ) but sparse matrix.Ground-state energies and mean interparticle distances for several confinement radii R are given in table 1.The pressure acting on the confined helium atom is also given in table 1.It is a e-mail: jdoheter@triumf.cab e-mail: dbaye@ulb.ac.beTable 1.Ground-state energy and mean interparticle distances of a helium atom confined in a sphere of radius R and pressure acting on such a confined helium atom [1].Atomic units are used.The powers of ten are indicated between brackets.calculated from the radius dependence of the energies E(R) by the formula [2][3][4],

R E r
where the derivative is evaluated by a finite-difference formula.A similar accuracy is obtained for the energy and interparticle distances of the first excited singlet level and the lowest triplet level [1].
Although the Lagrange-mesh method can be applied easily for small and large radii, it is particularly efficient for small radii (smaller variational basis size and better accuracy).The dependence of the ground-state energy on the confinement radius R is shown graphically in Fig. 1.For large confinement radii (R 2 fm), the confined helium energy tends nearly exponentially to the free helium energy.A parametrization of this curve could allow an easy evaluation of the pressure at very large radii, difficult to reach by direct calculations.

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DOI: 10.1051/ C Owned by the authors, published by EDP Sciences, 201

Figure 1 .
Figure 1.Difference between the ground-state energies of confined helium (E) and of free helium (E ∞ ) as a function of the confinement radius R. Atomic units are used.