8. Atoms
In the previous chapter, we considered atoms with a single electron (H, He+). The Hamiltonian for such atoms contains one kinetic-energy term and one potential-energy term. There are four internal degrees of freedom (
In this chapter, we move on to consider atoms with more than one electron. For these, we need an additional type of term into the Hamiltonian, the electrostatic repulsion energy between electrons.

Figure 8.1 Two electrons around a nucleus form a two-electron atom. The vectors
Take a pair of electrons, as shown in Figure 8.1. The electrostatis repulsion energy between the two is
where
Note
For a sense of scale: The electrostatic repulsion energy of two electrons that are 1 Å apart is +14.4 eV.
8.1. Hamiltonian and wavefunctions
Let us now set up the Hamiltonian for a two-electron atom (such as He or Li+). We need to include one kinetic and one electron-nuclear attraction potential energy term for each electron, and we also need one electron-electron repulsion term:
Each of the electrons has 4 degrees of freedom, making a total of 8. Therefore, the wavefunction describing the whereabouts of the two electrons depends on 8 variables.
(depending whether we choose Cartesian or spherical coordinates).
represents the probability of finding electron 1 with spin projection
To simplify notation, we introduce the following shorthand for the above wavefunction:
where 1 is an abbreviation for all coordinates of electron 1 (
In an atom with
Note
Consider the neutral Ne (neon) atom. It has 10 electrons,
8.2. Orbital approximation
In a multi-electron atom or molecule, the potential energy for one electron depends on the exact positions of all other electrons. Since electrons repel each other, they tend to avoid each other, and the positions of the electrons depend on each other: for example, if one electron is on the left, the other one tends to be on the right. This effect is called electron correlation, and it makes it impossible to solve the Schrödinger equation analytically.
To still obtain some insight, the following approximation is made: It is assumed that each electron moves in the average electrostatic field due to all the other electrons. This is called the mean-field approximation or orbital approximation.
Note
The orbital approximation is at the heart of chemical thinking and reasoning. It connects quantum theory with chemistry.
Figure 8.2 illustrates this for a two-electron atom. Within the orbital approximation, for the purpose of determining the whereabouts of electron 2, electron 1 is assumed delocalized. Repulsion is calculated not between electron 2 and electron 1, but between electron 2 and the static continuous charge distribution corresponding to the probability density of electron 1. In essence, the electrons are treated as if their positions were uncorrelated. Of course, this is a severe approximation, since this completely neglects electron correlation. In practice, the approximation is surprisingly useful.

Figure 8.2 The orbital approximation. Left: The instantaneous location of two electrons. Right: In the orbital approximation, electron 1 is modeled as a continuous charge distribution delocalized based on its orbital, and the repulsion energy of electron 2 from electron 1 is calculated as the average repulsion over the charge distribution of electron 1.
The main mathematical benefit of the orbital approximation is that the many-electron wavefunction can be written in terms of products of one-electron wavefunctions (orbitals), such as
Here, chemists say that “electron 1 is in orbital
Note
For example,
The effective repulsion potential energy function due to electron 1 acting on electron 2 is spherically symmetric, so that the spatial wavefunction for electron 2 can be approximated by
For atoms and molecules with more than 2 electrons, the reasoning is not quite as simple anymore.
8.3. Pauli principle and Slater determinants
The above product form of the 2-electron wavefunction is not quite physically correct. Electrons are indistinguishable, since all their intrinsic properties are identical (same charge, mass, spin, size). Therefore, labeling one electron as 1 and the other as 2 should give the same results as labeling the first one as 2 and the other one as 1. In other words, when two electron coordinates are exchanged in the wavefunction
The Hamiltonian is of course unaffected (
Note
When two electron coordinates are exchanged (swapped), the wavefunction
- The wavefunction is unchanged. It is called symmetric with respect to exchange. - The wavefunction changes sign. It is called antisymmetric with respect to exchange. - The wavefunction changes completely. It is neither symmetric nor antisymmetric with respect to exchange.
It turns out that, in order to be consistent with experimental observations, a many-electron wavefunction must change sign (= be antisymmetric) with respect to the exchange of the coordinates of any two electrons. This is the Pauli principle, or antisymmetrization postulate. For two electrons, it requires
For a valid three-electron wavefunction, the Pauli principles requires
More generally, a multi-particle wavefunction must be antisymmetric under the exchange of two identical fermions (electrons, protons, neutrons), and it must be symmetric under the exchange of two bosons (photons, 4He nuclei, etc.). Interestingly, all fermions have half-integer spin, and all bosons have integer spin.
Note
Let’s check whether the wavefunction for the electron configuration (1s
Exchanging the electron coordinates 1 and 2 gives the new wavefunction
This is identical to the starting wavefunction! This
This example illustrates another general principle that derives from the antisymmetrization postulate: The Pauli exclusion principle states that no two electrons can be in the same spinorbital - the orbitals must differ in at least one quantum number. This principle is the application of the more general Pauli principle to wavefunctions within the orbital approximation.
Let us examine the electron configuration (1s)2 = (1s
This wavefunction is antisymmetric under coordinate exchange and therefore represents a physical state. The sum of the two product functions gives a wavefunction that is symmetric under exchange, which is nonphysical.
A general recipe for constructing an antisymmetric multi-electron wavefunction with
(See the Appendix for details on determinants.) When expanded, this determinant gives a linear combination of
Note
The Slater determinant for the two-electron configuration (1s
which is exactly what we arrived at before, times the normalization constant
The Slater determinant wavefunction for 3 electrons in 3 orbitals is a linear combination of 3! = 6 products of the type
In chemistry, we often talk about “the electron in this orbital”. Although convenient, this is inaccurate language. Looking at the Slater determinant wavefunction, there is a term for every electron in every orbital! So we cannot assign specific electrons to specific orbitals - that is exactly the Pauli principle.
8.4. Variational theorem and method
With the orbital approximation, we have introduced an approximate, but more convenient, form for the multi-electron wavefunction
As a starting point, we take the expectation value of energy for an arbitrary unnormalized wavefunction
(The denominator is just the normalization integral.) According to the variational theorem,
In addition, only for

Figure 8.3 Illustrations of the variational theorem (left) and the variational method (right). The blue line indicates the true ground-state energy. The purple dots on the left indicate the energy expectation values for different trial wavefunctions. The purple curve on the right indicates the energy expectation value of a trial wavefunction that depends on a parameter
The variational method utilizes this theorem: Set up a trial wavefunction with adjustable parameters, e.g.
for
Note
Here is an example of the variational method, applied to the harmonic oscillator. Of course, for this the variational method is not needed as we know the exact ground-state energy and wavefunction. However, the example illustrates how the variational method works. Let’s take the trial function
where
Solving

8.5. Hartree-Fock and DFT
The Hartree-Fock method applies the variational method to an
These are called the Hartree-Fock equations, and the operator in (…) is called the Fock operator.
In order to have orbitals with variable shapes, a basis set expansion is used. Each orbital is represented as an adjustable linear combination of fixed basis functions.
Here
The basis functions are chosen to satisfy two criteria: (1) They should be similar in shape to the actual orbitals, so that the expansion can be short (i.e.
The

Figure 8.4 A Gaussian basis function representing the 1s orbital of a hydrogen atom. The function is a linear combination of three Gaussians. The exact wavefunction is shown as well.
In the variational method for determining the optimal wavefunction and energy for the ground state, the basis functions
There is an issue with solving the Hartree-Fock equations: The potential

Figure 8.5 The self-consistent field (SCF) method for solving the Hartree-Fock equations to obtain atomic and molecular orbitals.
Density functional theory (DFT) is a related variational method to calculate ground-state wavefunctions and energies. Like Hartree-Fock, it uses a Slater-determinant wavefunctions with orbitals, and it uses an SCF procedure. It differs from Hartree-Fock by adding semiempirical terms (so-called exchange-correlation functionals) to the Hamiltonian. These correction terms are designed to re-capture some of the electron-electron correlation effects that are neglected due to the use of a Slater determinant as the wavefunction. There is a large number of different exchange-correlation functionals that are referred to by acronyms, such as B3LYP, PBE, M06, etc. With the augmented Hamiltonian, DFT yields generally better results than Hartree-Fock. Hartree-Fock and DFT are just two methods of a large collection of ab initio methods (ab initio = from first principles) for solving the Schrödinger equation for atoms and molecules. Among these, DFT is by far the most broadly employed method.
8.6. Multi-electron atoms
Using the methods above, it is now possible to calculate the electronic structure of all atoms of the periodic table. The wavefunction for each atom is a Slater determinant built from orbitals that resemble the hydrogen orbitals in their angular shape, although they differ in their radial extent. The orbital energies
Whereas all states in the hydrogen atom with equal
The sequence of occupied orbitals in an atom is given by the Aufbau principle (“Aufbau” is German for “buildup” or “construction”): To obtain the ground-state electron configuration for an atom (or molecule) with

Figure 8.6 The sequence in which atomic orbitals are filled, according to the Madelung rule, indicated by the arrows starting on the top left. Note the similarity to the periodic table.
Note
A neutral carbon atom has 6 electrons. The Madelung rule predicts the ground-state electron configuration (1s)2(2s)2(2p)2.
Note
The rule has many exceptions. For example, consider Mn and Co 2+, both of which have 25 electrons. According to the rule, both should have the same electron configuration. Indeed, the experimentally determined electron configuration for Mn is [Ar]4s23d5, as predicted by the rule. However, for Co2+ it is [Ar]3d7 (not consistent with the rule). [Ar] indicates the electron configuration of argon (18 electrons): [Ar] = 1s2 2s22p63s23p6.
Besides dictating the sequence of filling with electrons, orbital energies have another specific use. According to the Koopmans theorem, the energy of the highest occupied orbital is approximately equal to the ionization energy ( = energy required to extract an electron from the atom). For example, if the orbital energy is -2.3 eV, then the ionization energy is approximately 2.3 eV. On the other hand, the energy of the lowest unoccupied orbital is approximately equal to the electron affinity, i.e. the energy released when an electron is added to an atom. Knowing these two energies for any atom (or molecule) allows the prediction of the direction of electron transfer, as illustrated in Figure 8.7.

Figure 8.7 Ionization energy (IE) and electron affinity (EA). Electron transfer is energetically favorable if the highest occupied orbital energy of one atom is higher than the lowest unoccupied orbital energy of the other.
One could be tempted to assume that the sum of all the occupied orbital energies is equal to the total energy of the atom. This is not the case. Summing the orbital energies double-counts the electron-electron repulsion energy and gives a value that is higher than the actual total energy. For example, in a two-electron atom, the orbital energies of both electron 1 and electron 2 include the 1-2 repulsion, which would therefore be double-counted.