Friday, February 28, 2014

Astatine: Halogen or Metal? Part 2: Introduction to Relativistic Quantum Chemistry

Click here for part 1, "Astatine: Halogen or Metal? Part 1: Background."

Note to readers: To date I have attempted to keep my posts approachable by non-chemists in an effort to provide chemical education to anyone interested. Due to the subject nature, this post will perforce be something of an exception and will contain parts that may be unaccessible to anyone without at least bachelor's level expertise in chemistry or physics. Nonetheless, I will attempt to make the conclusions accessible to a more general audience.

The Schrödinger equation is superb at describing things moving slowly when compared to the speed of light, and has made some truly astounding predictions in the century since its discovery. Unfortunately, it will begin to break down for objects moving at relativistic speeds. Indeed, the more ambitious readers can easily verify that the Schrödinger equation is not Lorentz invariant (is Lorentz variant?), and thus its solutions will not be true for all reference frames. There are a few relativistic equivalents to the Schrödinger equation, the one that's important to chemists is the Dirac equation.

A full solution for relativistic quantum chemistry would involve using quantum electrodynamics (QED) to find solutions to the Dirac equation, using these to create molecular orbitals, and building relativistic quantum chemistry from there. There is a known relativistic solution to the hydrogen atom known as the "Dirac atom" discovered in 1928 by Walter Gordon (1893-1939) and Charles Darwin (1887-1962, the grandson of the author of On the Origin of Species). However, for various reasons attempts to use the Dirac atom's solutions to create molecular orbitals for other atoms and compounds have not been fruitful (at least attempts for electronic structure models short of computationally expensive CI calculations).

Instead, we generally start with the standard Fock operator, and introduce a Dirac operator on top of that. So called Dirac-Hartree-Fock (DHF) can be extended to configuration interactions, coupled-cluster, and the like. An analogous approach to creating relativistic density functional theory (DFT) also exists, in which the relativistic 4-current replaces the electron density.

So when do we need to use relativistic quantum chemistry? There are three primary cases where quantum effects can become relevant in quantum chemistry. They are, in order of decreasing significance, (1) s orbital shrinkage due to inner electron velocity, (2) spin-orbit interactions, and (3) vacuum polarization.

(1) s Orbital Shrinkage

Orbital size will be influenced by the velocity of the particles under study. Due to the statistical nature of quantum mechanics, we cannot properly speak of the "velocity" of electrons moving about a nucleus--they exist as delocalized wave functions before they're observed anyway. What we can speak of is the average speed, which is in line with a statistical understanding. The specific metric we will use is the root-mean-square (RMS) speed. It can be shown (see, for example, Quantum Chemistry by Levine section 16.7) that the RMS speed of the 1s electrons is approximately equal to Zc137, where c is the speed of light and Z is the nuclear charge (i.e. number of protons).

We can get a general idea of the effect of the above by performing a "back of the envelope calculation" as follows: For astatine we calculate a RMS for the 1s electrons of 62% the speed of light, which in turn results in a mass 128% the rest mass. Since the 1s orbital's radius is inversely proportional to the electron mass, the 1s orbital's radius shrinks to roughly 78% its non-relativistic value. Although electrons in farther s orbitals do not have nearly as high an RMS, in order to maintain their orthogonality to the 1s orbital they will also contract. The p orbitals will also exhibit some shrinkage, while the d and f orbitals will actually expand due to the increased nuclear shielding by the inner orbitals.

The above will result in relativistic effects having an approximately linear dependence on Z. Other more advanced treatments have proposed that they may actually depend on Z2, or even some higher power such as Z4. In any event, the high speed on the inner electrons and consequent change in orbital size is usually considered the most dominant result of relativistic quantum mechanics. It generally results in more stable and chemically inert orbitals, and shorter than expected bond-lengths in compounds.

(2) Spin-Orbit Interactions

A second relativistic effect is spin-orbit coupling. This is usually considered to by of lesser importance than orbital shrinkage, although cases can arise in which it will be the dominant factor.

A full treatment of spin-orbit coupling would start with the Dirac equation, but we can understand spin-orbit coupling qualitatively as follows: Imagine we are riding an electron around a nucleus. We will observe the nucleus moving around us. As the nucleus is a moving electric charge, it will create a magnetic field, which will interact with intrinsic spin magnetic moment of the electron.

The higher the orbital angular momentum of the electron, the faster it will observe the nucleus moving, and hence the stronger the interaction between the two. This is responsible, for example, for the fine structure splitting of the sodium D line by raising the p3/2 level to a higher energy level than the p1/2 level.

For lighter atoms, spin-orbit coupling can essentially be viewed as a perturbation on the standard Hamiltonian (hydrogen orbitals plus electron repulsion) in a scheme known as Russell-Saunders coupling (or LS coupling). For heavier elements, spin-orbit interactions become comparable in magnitude to or even greater than inter-electron repulsions. In such a case we can instead create a Hamiltonian from the hydrogenic orbitals plus the spin-orbit coupling, and then treat inter-electron repulsions as the perturbation. This scheme is known as j-j coupling. However, most heavy atoms are actually somewhere intermediate between Russell-Saunders coupling and j-j coupling, substantially complicating matters.

(3) Vacuum Polarization

Another relativistic effect worth mentioning in passing is vacuum polarization. According to QED, the vacuum itself continuously creates and annihilates virtual electron-positron pairs. For a pair created between a nucleus and an electron, the virtual electron will tend toward the nucleus, while the virtual positron will tend toward the electron. This results in the vacuum itself behaving as a dielectric medium, and screens the charge felt by the (real) electron. When an electron is extremely close to the nucleus, it will feel a higher charge than the normally measured one, otherwise it will feel the screened charge. This results in a few observable results, most notably the Lamb shift in the hydrogen spectrum, but is generally not important for quantum chemistry results for outer shell electrons.

Note that pair creation-annihilation is not included in the Dirac equation, and must be added as an addition calculation.

Now that we have explained the basic problems that prevent us from using standard electronic structure techniques on heavy atoms, we are ready to discuss relativistic electronic structure calculations on astatine. Coming soon, Click here for the final installment, Astatine: Halogen or Metal? Part 3: Electronic Structure Calculations.