1 Introduction
1.8 Quantum Mechanics
1.8.1 A hydrogen-like atom
A quantum analogue of the Sun–Earth system is a nucleus and one electron, i.e. a hydrogen-like atom. The force holding the nucleus and electron together is the Coulomb interaction.
Figure 1.5 A Hartree–Fock model for the solar system
Figure 1.6 Modelling the solar system with actual interactions
(1.31) The interaction again only depends on the distance, but owing to the small mass of the electron, Newton’s equation must be replaced with the Schrödinger equation. For bound states, the time-dependence can be separated out, as shown in Section 1.6.1, giving the time-independent Schrödinger equation.
(1.32) The Hamiltonian operator for a hydrogen-like atom (nuclear charge of Z) can in Cartesian coordinates and atomic units be written as eq. (1.33), with M being the nuclear and mthe electron mass (m=1 in atomic units).
(1.33) The Laplace operator is given by eq. (1.34).
(1.34) The two kinetic energy operators are already separated, since each only depends on three coordinates. The potential energy operator, however, involves all six coordinates.
The centre of mass system is introduced by the following six coordinates.
(1.35)
Here the X,Y,Zcoordinates define the centre of mass system, and the x,y,zcoordi- nates specify the relative position of the two particles. In these coordinates the Hamil- tonian operator can be rewritten as eq. (1.36).
(1.36) The first term only involves the X,Yand Zcoordinates, and the ∇2XYZoperator is obvi- ously separable in terms of X,Yand Z. Solution of the XYZpart gives translation of the whole system in three dimensions relative to the laboratory-fixed coordinate system. The xyzcoordinates describe the relative motion of the two particles in terms of a pseudo-particle with a reduced mass mrelative to the centre of mass.
(1.37) m=
+ =
(
+)
≅M m
M m
m m
M
nuc elec m
nuc elec
elec elec
nuc
elec
1 H= − ∇ − ∇ −
+ +
1 2
2 2
2 2 2
1
XYZ 2 xyz
Z x y z m
X Mx mx
M m x x x
Y My my
M m y y y
Z Mz mz
M m z z z
=( + )
( + ) = −
=( + )
( + ) = −
=( + )
( + ) = −
1 2
1 2
1 2
1 2
1 2
1 2
;
;
;
∇ =i + +
i i i
x y z
2 2
2 2
2 2
2
∂
∂
∂
∂
∂
∂ H= − ∇ − ∇ −
( − ) +( − ) +( − ) 1
2
1
12 2
2 2
1 2
2
1 2
2
1 2
M m 2
Z
x x y y z z
HΨ=EΨ V r12 1 2
12
( )= q q r
The potential energy becomes very simple, but the kinetic energy operator becomes complicated.
(1.38)
The kinetic energy operator, however, is almost separable in spherical polar coordinates, and the actual method of solving the differential equation can be found in a number of textbooks. The bound solutions (negative total energy) are called orbitals and can be classified in terms of three quantum numbers,n,land m, corresponding to the three spatial variables r,qand j. The quantum numbers arise from the boundary conditions on the wave function, i.e. it must be periodic in the qand jvariables, and must decay to zero as r→ ∞. Since the Schrödinger equation is not completely separable in spherical polar coordinates, there exist the restrictions n>l≥ |m|.The nquantum number describes the sizeof the orbital, the lquantum number describes the shapeof the orbital, while the mquantum number describes the orientationof the orbital relative to a fixed coordinate system. The lquantum number translates into names for the orbitals:
• l=0 : s-orbital
• l=1 : p-orbital
• l=2 : d-orbital, etc.
The orbitals can be written as a product of a radialfunction, describing the behaviour in terms of the distance rbetween the nucleus and electron, and spherical harmonic functions Ylmrepresenting the angular part in terms of the angles qand j. The orbitals can be visualized by plotting three-dimensional objects corresponding to the wave function having a specific value, e.g.Ψ2=0.10.
H= − ∇ −
∇ = + +
1 2
1 1 1
2
2 2
2
2 2 2
2 2
m
∂
∂
∂
∂ q
∂
∂q q ∂
∂q q
∂
∂j
qj
qj
r
r
Z r r rr
r r sin sin r sin
r
ϕ θ
x y z
x = rsinθcosϕ y = rsinθsinϕ z= rcosθ
Figure 1.7 A spherical polar coordinate system
For the hydrogen atom, the nucleus is approximately 1800 times heavier than the electron, giving a reduced mass of 0.9995melec. Similar to the Sun–Earth system, the hydrogen atom can therefore to a good approximation be considered as an electron moving around a stationary nucleus, and for heavier elements the approximation becomes better (with a uranium nucleus, for example, the nucleus/electron mass ratio is ~430 000). Setting the reduced mass equal to the electron mass corresponds to making the assumption that the nucleus is infinitely heavy and therefore stationary.
The potential energy again only depends on the distance between the two particles, but in contrast to the Sun–Earth system, the motion occurs in three dimensions, and it is therefore advantageous to transform the Schrödinger equation into a spherical polar set of coordinates.
The orbitals for different quantum numbers are orthogonal and can be chosen to be normalized.
(1.39) The orthogonality of the orbitals in the angular part (l and m quantum numbers) follows from the shape of the spherical harmonic functions, as these have lnodal planes (points where the wave function is zero). The orthogonality in the radial part (n quantum number) is due to the presence of (n–l–1) radial nodes in the wave function.
In contrast to classical mechanics, where all energies are allowed, wave functions and associated energies are quantized, i.e. only certain values are allowed. The energy only depends on nfor a given nuclear charge Z, and is given by eq. (1.40).
(1.40) Unbound solutions have a positive total energy and correspond to scattering of an electron by the nucleus.
1.8.2 The helium atom
Like the solar system, it is not possible to find a set of coordinates where the Schrödinger equation can be solved analytically for more than two particles (i.e. for many-electron atoms). Owing to the dominance of the Sun’s gravitational field, a central field approximation provides a good description of the actual solar system, but this is not the case for an atomic system. The main differences between the solar system and an atom such as helium are:
(1) The interaction between the electrons is only a factor of two smaller than between the nucleus and electrons, compared with a ratio of at least 1000 for the solar system.
(2) The electron–electron interaction is repulsive, compared with the attraction between planets.
E Z
= − n22 2
Ψn l m, , Ψn l m′ ′ ′, , =dn n, ′d dl l,′ m m, ′
Table 1.2 Hydrogenic orbitals obtained from solving the Schrödinger equation
n l m Ψn,l,m(r,q,j) Shape and size
1 0 0 Y0,0(q,j)e−Zr
2 0 0 Y0,0(q,j)(2 −Zr)e−Zr/2 1 ±1, 0 Y1,m(q,j)Zre−Zr/2
3 0 0 Y0,0(q,j)(27 −18Zr+2Z2r2)e−Zr/3 1 ±1, 0 Y1,m(q,j)Zr(6 −Zr)e−Zr/3 2 ±2,±1, 0 Y2,m(q,j)Z2r2e−Zr/3
(3) The motion of the electrons must be described by quantum mechanics owing to the small electron mass, and the particle position is determined by an orbital, the square of which gives the probability of finding the electron at a given position.
(4) Electrons are indistinguishable particles having a spin of 1/2. This fermion charac- ter requires the total wave function to be antisymmetric, i.e. it must change sign when interchanging two electrons. The antisymmetry results in the so-called exchange energy, which is a non-classical correction to the Coulomb interaction.
The simplest atomic model would be to neglect the electron–electron interaction, and only take the nucleus–electron attraction into account. In this model each orbital for the helium atom is determined by solving a hydrogen-like system with a nucleus and one electron, yielding hydrogen-like orbitals, 1s, 2s, 2p, 3s, 3p, 3d, etc., with Z=2. The total wave function is obtained from the resulting orbitals subject to the aufbau and Pauli principles. These principles say that the lowest energy orbitals should be filled first and only two electrons (with different spin) can occupy each orbital, i.e. the elec- tron configuration becomes 1s2. The antisymmetry condition is conveniently fulfilled by writing the total wave function as a Slater determinant, since interchanging any two rows or columns changes the sign of the determinant. For a helium atom, this would give the following (unnormalized) wave function, with the orbitals given in Table 1.2 with Z=2.
(1.41) The total energy calculated by this wave function is simply twice the orbital energy,
−4.000 au, which is in error by 38% compared with the experimental value of −2.904 au. Alternatively, we can use the wave function given by eq. (1.41), but include the electron–electron interaction in the energy calculation, giving a value of −2.750 au.
A better approximation can be obtained by taking the averagerepulsion between the electrons into account when determining the orbitals, a procedure known as the Hartree–Fock approximation. If the orbital for one of the electrons were somehow known, the orbital for the second electron could be calculated in the electric field of the nucleus and the first electron, described by its orbital. This argument could just as well be used for the secondelectron with respect to the firstelectron.The goal is therefore to calculate a set of self-consistent orbitals, and this can be done by iterative methods.
For the solar system, the non-crossing of the planetary orbitals makes the Hartree–Fock approximation only a very minor improvement over a central field model. For a many-electron atom, however, the situation is different since the position of the electrons is described by three-dimensional probability functions (square of the orbitals), i.e. the electron “orbits” “cross”. The average nucleus–electron distance for an electron in a 2s-orbital is larger than for one in a 1s-orbital, but there is a finite probability that a 2s-electron is closer to the nucleus than a 1s-electron. If the 1s- electrons in lithium were completely inside the 2s-orbital, the latter would experience an effective nuclear charge of 1.00, but owing to the 2s-electron penetrating the 1s- orbital, the effective nuclear charge for an electron in a 2s-orbital is 1.26. The 2s-elec- tron in return screens the nuclear charge felt by the 1s-electrons, making the effective nuclear charge felt by the 1s-electrons less than 3.00. The mutual screening of the two 1s-electrons in helium produces an effective nuclear charge of 1.69, yielding a total
Φ = ( ) ( )
( )
( )= ( ) ( )− ( ) ( ) f
f
f
f f f f f
a a
b b
a b b a
1 1
1 1
1 1 1 1
1 2
1
2 1 2 1 2
s s
s s
s s s s
The equal mass of all the electrons and the strong interaction between them makes the Hartree–Fock model less accurate than desirable, but it is still a big improvement over an independent orbital model. The Hartree–Fock model typically accounts for
~99% of the total energy, but the remaining correlation energyis usually very impor- tant for chemical purposes. The correlation between the electrons describes the
“wiggles” relative to the Hartree–Fock orbitals due to the instantaneous interaction between the electrons, rather than just the average repulsion. The goal of correlated methods for solving the Schrödinger equation is to calculate the remaining correction due to the electron–electron interaction.