Quantum Mechanics does not predict Atomic and Molecular Orbitals
Theoretical atomic, molecular and condensed-matter physicists and also theoretical chemists make proper use of orbitals and related techniques. They are aware that such concepts, while useful, can be potentially misleading if taken naively. However in my teaching experience I noticed that many people mistake the orbitals for something fundamental. This note is a non-technical introduction to the reasons why caution is needed and a hint to the correct understanding of many-electron wave functions.
Early Quantum Mechanics and the H atom
What is the ground state electron wave function in the H atom? Most introductory texts in Quantum Mechanics say that it is the 1s orbital,which is of the form
where r is the radial coordinate of the electron,
is the Bohr radius,
the electron mass,
h is Planck’s constant, and e the electron charge. This orbital is obtained by solving the stationary state Schrödinger equation
The Hamiltonian operator reads:
and the second term V(r) is the Coulomb interaction. This model of the H atom was solved using different methods by Pauli in 1925 and by Schrödinger in 1926 and it was immediately clear that the (long sought after) breakthrough towards Quantum Theory had occurred. The simple wave function of Equation (1) characterized the lowest orbital of Hydrogen, which is known as 1s level for historical reasons related to early spectroscopic discoveries. However we also need to account for the electron spin and also to include the relativistic spin-orbit interaction in excited states having angular momentum; thus one must really speak about spin-orbitals.
The excited bound states of Hydrogen can be assigned to orbitals having (among other quantum numbers) a higher integer principal quantum numbers n . Moreover the concept of spin-orbital has been useful in the study of the heavier atoms, with several electrons; the simplest description uses the so called Central Field Approximation. Indeed each atomic electron is acted upon by the potential of the nucleus and of the other electrons, and the overall potential has a nearly spherical symmetry. In the central field approximation all atoms have been assigned electronic configurations (but for heavy atoms a relativistic treatment is essential.) From the electronic configurations one can infer some chemical properties of the elements. Then, the orbital related concepts have been extended to molecules and condensed matter as well. In short, the fact that such ideas are useful is out of question; in this note however I would like to emphasize that if one ‘believes’ too much in the idea of an orbital, one mistakes a convenient simplification for reality and may come to wrong conclusions.
Even the early result (1) should not be taken more literally than it deserves. It neglects the nuclear and electron spins and their interactions, the nuclear size, all relativistic effects, which are included in Dirac’s theory, retardation and other effects of Quantum Electrodynamics, like the Lamb shift. However this note is not intended to show how to obtain better orbitals; I wish to show that in principle no electron orbital (or spin-orbital) really exists.
The Two-body Problem in Classical Mechanics
Actually, the electron is not moving in a fixed potential well; it is bound to a moving proton instead; so one must solve a two-body problem. For clarity, I start from Classical Mechanics. The total kinetic energy is
in obvious notation. Thus, one should start with a more realistic Hamiltonian compared to (2), namely,
including the kinetic energy of the proton explicitly with the inter-particle separation
Equation (3) holds in any inertial system; our next task is the separation of the kinetic energy due to the motion of the center of mass of the atom from the kinetic energy in the center of mass reference system. One exploits the fact that the atom is an isolated system and therefore its total momentum
is conserved, The translational kinetic energy of the motion of the center of mass of the atom as a whole is (again, in obvious notation)
here I have introduced the total momentum and the total mass. The center of mass of the atom given by:
a center of mass reference, where
is such that (6) and (7) vanish. Using (5) one finds
Hence the difference
is the kinetic energy in the center-of-mass system. Actually, K has a simple expression. Setting
one finds:
then, introducing the reduced mass,
and the momentum associated to the relative motion
one finds the kinetic energy in the center-of-mass system
Putting together (3),(6), (7) and (14) one finds that in any inertial system:
The first term is the translation kinetic energy (7), while the rest is the energy of the relative motion. This is a two-body Hamiltonian, but in the center-of-mass system, the first term vanishes, and
becomes an effective one-body problem for the relative motion. In this way, one determines the individual orbits of the electron and of the proton, similar to those characteristic of celestial mechanics. The next is an important remark: in order to determine the motion in any inertial reference system, once
is assigned and the motion
in the center-of-mass system is found, one just solves the system of equations (5) and (8). In the same way, in the Kepler problem, one can determine the motion of both the star and a planet as seen from the Earth.
The same Two-body Problem in Quantum Mechanics
In this note I will limit myself to the non-relativistic theory, to avoid excessive complications, but the arguments presented here also extend to the relativistic case. Working out the problem in Quantum Theory, one can repeat all the same transformations up to Equation (15); the two-body Hamiltonian comes in the same form, but then suddenly all the differences between classical and quantum become dramatic. Now we must solve the two-particle Schrödinger equation for the stationary states,
and the wave function is a two-particle wave function. One finds
where the first factor is a plane wave function which represents the free motion of the center of mass. In the rest system of the atom, the first factor is a constant, and
which looks like the orbital of Equation (1), except that the electron mass is replaced by the effective mass. However this is not a justification of (1) as an orbital for an electron. The mass is different and (more importantly) Equation (19) depends on the coordinates of both proton and electron. Actually, it does not say anything about the probability amplitude of finding the electron at a given place, so it is no orbital at all! The quantum and classical descriptions suddenly disagree. The disagreement is conceptual. While classically we could determine the individual motions of both particles, there is no way to break a two-body amplitude into one-particle wave functions. The states of electron and nucleus are correlated and entangled! Entanglement implies that one knows the state of the quantum system without knowing the state of its parts. This is weird, because of course classically the opposite is true.
In the case of Hydrogen, one might argue that mistaking (1) for an electron wave function is not too bad, since after all the proton mass is about 1800 times the electron mass, presumably the proton spends most time close to the origin and the effective mass is not so different from m; but please note, in Atomic Physics the measurements are extremely precise, and small differences matter quite a lot. Moreover there are several interesting Hydrogen-like cases in which the mass of the positive charge is not overwhelming. In muonic atoms the electron is replaced by the 200 times heavier muon. In Positronium the mass of the positron is the same as the mass of the electron, and there are other such cases of interest (Quarkonium is a bound state of a quark and an anti-quark, and at short range the interaction is mainly Coulombian). In such cases, the two-body problem cannot be thought of in terms of one-particle wave functions. For this reason, even in ordinary Hydrogen, the idea of an atomic orbital, while useful, must be taken as an approximation to reality. For instance, hydrogen bonds in water and other solvents remind us that actually the proton is indeed entangled with the electrons.
Considerations on the use of electron Spin-Orbitals in Atomic Physics and in Many Body Quantum Mechanics
In this short note I can’t even begin to talk about the modern quantum theory of atoms, molecules and solids; it is an involved, specialist topic that requires years of study. However, the above argument suggests that if we could solve the Schrödinger equation with a N body Hamiltonian exactly, we should find many body wave functions, but no orbitals at all. In the above case with 2 particles, the solution was easy because the separation of the center of mass motion left us with an effective one-body problem, but in the case with N particles, after the separation of the center of mass motion, no fewer than N-1 remain entangled. For Helium, N=3, we are left with an effective two-body one. For Uranium the effective problem with 93 bodies (considering the 92 electrons and the nucleus) is not greatly reduced in dimensions by introducing the effective mass. This proliferation of dimensions is a terrific complication. Every particle is entangled with every other particle. It is the usual property of quantum systems: you may know the state of the system, but that does not tell the state of its parts. As a result, one should work in a space with a huge number of dimensions, and the problem quickly explodes with increasing N. Individual spin-orbitals for each particle would be very much more tractable, but above we reached the general conclusion is that one cannot speak about an electron orbital whenever the electron (or the nucleus) interacts with some other microscopic entity. One needs a better idea before dealing with molecules and condensed matter problems.
Therefore, the pragmatic way of setting the many-body problem, actually the only way, is to pretend, at least at the beginning, that all the particles do not interact with each other, but instead interact with an effective potential that allows to justify the use of spin-orbitals. The hope is to be able to correct this model afterwards, by cleverly introducing back the complications that do not allow the orbitals to really exist. Well, this is the winning strategy, even beyond hope.
The nuclei are heavy and in almost all problems, the effects of their motions can be included as a perturbation, by introducing suitable electron-phonon or electron-vibration interactions. In Solid State Physics, the theory of Polarons and the standard theory of Superconductivity work in this way. Then, by far the most important concern is with electron-electron interactions. Since the electrons are Fermions, the many-electron wave function must be antisymmetric in the exchange of two electrons (by the Pauli spin-statistic theorem). Therefore, the simplest approximation to the electronic state is in terms of Slater determinants, made with spin-orbitals; in this way the Pauli principle is built in automatically. Since the spin-orbitals make a complete set, by mixing enough of them one can solve the problem of electron-electron electromagnetic interactions exactly. The direct expansion in determinants is called Configuration Mixing.
The simplest representation of the electronic state is obtained variationally in the Hartree-Fock approximation, or Dirac-Fock in the relativistic case) and any state with the correct exchange symmetry can be obtained by superimposing determinants from a complete set. A complete set of spin-orbitals is needed to set up second quantization and Green’s function methods. The functional density method introduced by W. Kohn and coworkers also allows to calculate determinantal states keeping correlation into account exactly in principle, but to some extent in practice. In atomic physics, each determinant corresponds to an electronic configuration; however, the electron-electron interaction mixes different configurations and in general an infinite expansion is needed. Field theoretical methods are commonly used today. This topic is well outside my scope here; some many body approaches are presented in several books including one of mine [1] .
Life is not always hard. Historically, the first attempted applications of Quantum Mechanics to the study of solids led to unexpected, easy success. Calculating the propagation of a single electron in a Silicon, Sodium,Graphite, Gold or Aluminum crystal it was possible to explain most properties of these ‘simple’ metals and semiconductors. Spin-orbitals belonging to states below the Fermi levels are occupied at absolute zero, those above are empty. In the thirties nobody knew how to deal with electron-electron interactions, yet in many cases one could forget about them, calculate the band structure as if ‘the electron’ propagated in a periodic potential and derive the band structure.
The same simple approaches failed completely in other cases, such as magnetic materials. At first the situation was hard to understand, but then Lev Landau explained the simple cases with his quasi-particle theory, which explains why a simple-minded approach may succeed in some cases. Similar comments apply to the partial success of he shell model in nuclear Physics.
Sometimes one needs qualitative insights on the properties of a series of similar molecules, like for instance conjugated hydrocarbons. In fact, a clever choice of molecular orbitals, for instance on the LCAO method (where LCAO stands for Linear Combination of Atomic Orbitals) leads in a relatively very cheap, but qualitatively correct predictions about absorption spectra and other electronic properties. This is not too surprising since such calculations do contain some information about the system, its geometry, the atomic species involved, the number of electrons. This tends to credit the orbitals as true features of reality. Failures are easily forgiven and forgotten when a simple method provides useful results almost for free.
However it would be wrong to think that there are electrons that inhabit deep levels, while others are valence electrons. Moreover, it is tempting to relate the chemical properties directly to the shape of the valence orbitals, but such temptations may be misleading. Indeed, it is easy to show that the orbitals are not only mathematical constructions but indeed they are largely arbitrary.
How to change all orbitals without changing anything
In fact one can upset all the spin-orbitals without changing the determinant. Suppose we have a N electron state which is a determinant of spin-orbitals, namely,
with
where the index i runs over spin-orbitals and j over electrons. Now we may pick a N by N unitary matrix A; I recall that unitary means that
where the first factor is the Hermitean conjugate of the second and of course 1 stands for the unit matrix. Then the Binet theorem grants that
yields the same determinant, that is, the same wave function as M. In other terms, one can perform a unitary transformation thereby mixing all the spin-orbitals while the determinant, which is the quantum wave function, remains unaltered. This shows that while the many-electron determinant (or better, a many-determinant state) represents the Physics, at least in a Hartree-Fock approximation, the spin-orbitals have no direct physical meaning whatsoever.
In conclusion, orbitals are useful ingredients in the quantum mechanical formalism. However, they cannot be mistaken for something real, although they can have a limited predictive power in some classes of problems.
[1] Michele Cini, Topics and Methods in Condensed Matter Theory’, Springer Verlag (2007)
