Hydrogen-like atom: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Paul Wormer
imported>Paul Wormer
No edit summary
Line 1: Line 1:
A '''hydrogen-like atom'''  is an [[atom]] with one  [[electron]].  Except for the hydrogen atom itself (which is neutral) these atoms carry positive charge ''e(Z-1)'', where ''Z'' is the [[atomic number]] of the atom.  Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles,  their (non-relativistic) Schrödinger equation can be solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like [[atomic orbitals]]. These orbitals differ from one another in one respect only: the nuclear charge ''eZ'' appears in the radial part of the wave function.
In [[physics]] and [[chemistry]], a '''hydrogen-like atom''' (or ''hydrogenic'' atom) is an [[atom]] with one  [[electron]].  Except for the hydrogen atom itself (which is neutral) these atoms carry positive charge ''e(Z-1)'', where ''Z'' is the [[atomic number]] of the atom and ''e'' is the [[elementary charge]].  Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles,  their non-relativistic [[Schrödinger equation]] can be solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like [[atomic orbitals]]. These orbitals differ from one another in one respect only: the nuclear charge ''eZ'' appears in the radial part of the wave function.


==Quantum numbers==
==Quantum numbers==

Revision as of 08:04, 17 September 2007

In physics and chemistry, a hydrogen-like atom (or hydrogenic atom) is an atom with one electron. Except for the hydrogen atom itself (which is neutral) these atoms carry positive charge e(Z-1), where Z is the atomic number of the atom and e is the elementary charge. Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles, their non-relativistic Schrödinger equation can be solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals. These orbitals differ from one another in one respect only: the nuclear charge eZ appears in the radial part of the wave function.

Quantum numbers

Hydrogen-like atomic orbitals are eigenfunctions of a Hamiltonian (energy operator) with eigenvalues proportional to 1/n², where n is a positive integer. Further the obitals are usually chosen such that they are also eigenfunctions of the square of the one-electron angular momentum vector operator l ≡ (lx, ly, lz) and of lz. The operator l 2lx2 + ly2 + lz2 has an eigenvalue proportional to l(l+1), where l is a non-negative integer. The operator lz has an eigenvalue proportional to an integer usually denoted by m.

Note that the energies of the hydrogen-like orbitals do not depend on the angular momentum quantum numbers l and m, but solely on the principal quantum number n. The degeneracy (maximum number of linearly independent eigenfunctions of same energy) of energy level n is equal to n2. This is the dimension of the irreducible representations of the symmetry group of hydrogen-like atoms, which is SO(4).

A hydrogen-like atomic orbital is uniquely identified by the values of the principal quantum number n, the angular momentum quantum number l, and the magnetic quantum number m. These quantum numbers are integers and we summarize their ranges:

To this must be added the two-valued spin quantum number ms = ±½ in application of the exclusion principle. This principle restricts the allowed values of the four quantum numbers in electron configurations of more-electron atoms: it is forbidden that two electrons have the same four quantum numbers.

Completeness of hydrogen-like orbitals

In quantum chemical calculations hydrogen-like atomic orbitals cannot serve as an expansion basis, because they are not complete. The non-square-integrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space.[1]

Schrödinger equation

The atomic orbitals of hydrogen-like atoms are solutions of the time-independent Schrödinger equation in a potential given by Coulomb's law:

where

After writing the wave function as a product of functions:

(in spherical coordinates), where are spherical harmonics, we arrive at the following Schrödinger equation:

where μ is, approximately, the mass of the electron. More accurately, it is the reduced mass of the system consisting of the electron and the nucleus. Because the electron mass is about 1836 smaller than the mass of the lightest nucleus (the proton), the value of μ is very close to the mass of the electron me for all hydrogenic atoms. In the remaining of the article we will often make the approximation μ = me. Since me will appear explicitly in the formulas it will be easy to correct for this approximation if necessary.


Wave function and energy

In addition to l and m, there arises a third integer n > 0 from the boundary conditions imposed on R(r). The final expression for the normalized wave function is:

where:

  • are the generalized Laguerre polynomials in the definition given here.
Note that is approximately equal to (the Bohr radius). If the mass of the nucleus is infinite then and .
  • . (Energy eigenvalues. As we pointed out above they depend only on n, not on l or m).
  • function is a spherical harmonic.

Solution

The kinetic energy operator in spherical polar coordinates is

The spherical harmonics satisfy

.

Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,

Note that the first term in the kinetic energy can be rewritten

.

This follows because both sides of this equation can be shown by application of the product rule to be equal to a third form of this operator:

If subsequently the substitution is made into

the radial equation becomes

which is precisely a Schrödinger equation for the function u(r) with an effective potential given by

where the radial coordinate r ranges from 0 to . The correction to the potential V(r) is called the centrifugal barrier term.

In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length, respectively,

.

Substitute and into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,

Two classes of solutions of this equation exist: (i) W is negative, the corresponding eigenfunctions are square integrable and the values of W are quantized (discrete spectrum). (ii) W is non-negative. Every real non-negative value of W is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable. In the remaining part of this article only class (i) solutions will be considered. The wavefunctions are known as bound states, in contrast to the class (ii) solutions that are known as scattering states.

For negative W the quantity is real and positive. The scaling of y, i.e., substitution of gives the Schrödinger equation:

For the inverse powers of x are negligible and a solution for large x is . The other solution, , is physically non-acceptable. For the inverse square power dominates and a solution for small x is xl+1. The other solution, x-l, is physically non-acceptable. Hence, to obtain a full range solution we substitute

The equation for fl(x) becomes,

Provided is a non-negative integer, say k, this equation has polynomial solutions written as

which are generalized Laguerre polynomials of order k. We will take the convention for generalized Laguerre polynomials of Abramowitz and Stegun.[2] Note that the Laguerre polynomials given in many quantum mechanical textbooks, for instance the book of Messiah,[3] are those of Abramowitz and Stegun multiplied by a factor (2l+1+k)! The definition given in this article coincides with the one of Abramowitz and Stegun.

The energy becomes

The principal quantum number n satisfies , or . Since , the total radial wavefunction is

with normalization constant

which belongs to the energy

In the computation of the normalization constant use was made of the integral [4]

References

  1. This was observed as early as 1929 by E. A. Hylleraas, Z. f. Physik vol. 48, p. 469 (1929). English translation in H. Hettema, Quantum Chemistry, Classic Scientific Papers, p. 81, World Scientific, Singapore (2000). Later it was pointed out again by H. Shull and P.-O. Löwdin, J. Chem. Phys. vol. 23, p. 1362 (1955).
  2. Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4.
  3. A. Messiah, Quantum Mechanics, vol. I, p. 78, North Holland Publishing Company, Amsterdam (1967). Translation from the French by G.M. Temmer
  4. H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, Van Nostrand, 2nd edition (1956), p. 130. Note that convention of the Laguerre polynomial in this book differs from the present one. If we indicate the Laguerre in the definition of Margenau and Murphy with a bar on top, we have .