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Chemical Table of the Elements

The rules governing the one-electron atom wavefunction Wnii!m>m and the Pauli exclusion principle, which states that only one electron can be accommodated in a completely described quantum state, are the basis for the Chemical Table of the Elements. As we have seen, the strange rules of nanophysics allow In2 distinct states for each value of the principal quantum number, n. There are several notations to describe this situation. The "K shell" of an atom comprises the two electrons of n = l(ls2), followed by the "L shell" with n = 2(2s22p6); and the "M shell" with n = 3 (3s2 3p6 3d10). These closed shells contain, respectively, 2, 8, and 18 electrons.

In the Chemical Table of Mendeleyev, one notable feature is the stability of filled "electron cores", such as those which occur at Z = 2 (He, with a filled K shell), and 2 = 10 (Ne, with filled K and L shells).

The situation of a single electron beyond a full shell configuration, such as sodium, potassium, rubidium, and cesium, can be roughly modeled as an ns electron moving around the rare gas core described with an effective charge 7!, less than 2. That 2' is reduced results from the shielding of the full nuclear charge 2 by the inner closed shell electrons. It is remarkable that interactions between electrons in large atoms can be in many cases ignored.

These rules of nanophysics are believed to account for the schematics of the chemical table of the elements, and, as well, to the properties of chemical compounds. It is a logical progression to expect that the larger aggregations of molecules that are characteristic of biology are also understandable in the framework of Schro-dinger quantum mechanics.

Nano-symmetry, Di-atoms, and Ferromagnets

A further and profound nanophysical behavior, with large consequences for the macroworld, is based on the identical nature of the elementary particles, such as electrons.

It is easy to understand that ionic solids like KC1 are held together by electrostatic forces, as the outer electron of K fills the outer electron shell of CI, leading to an electrostatic bond between K+ and Cl~.

But what about the diatomic gases of the atmosphere, H2, 02, and N2? The bonding of these symmetric structures, called covalent, is entirely nanophysical in its origin. It is strange but true that the same symmetry-driven electrostatic force that binds these di-atoms is also involved in the spontaneous magnetism of iron, cobalt and other ferromagnetic metals.

Indistinguishable Particles, and their Exchange

The origin of the symmetric covalent bond is the indistinguishable nature of electrons, for which labels are impossible. If two electrons are present in a system, the probability distributions P(x\,x2) and P(x2,x1) must be identical.

No observable change can occur from exchanging the two electrons. That is,

from which it follows that either

V»,m(*27*i) = ipn,m(xi>%2) (symmetric case), or (5.2)

tyn,m(x2>xi):=-i>n,m{xi>x2) (antisymmetric case). (5.3)

If we apply this idea to the two non-interacting electrons in, for example, a ID trap, with the wavefunction (4.39)

ipn,m(xi>x2)=A2 sin(njtXi/L) sm(mjix2/L) = ipn(xi) ipm(x2), we find that this particular two-particle wavefunction is neither symmetric nor antisymmetric. However, the combinations of symmetric and antisymmetric two-particle wavefunctions

ipA(l,2) = [ipn(x0 ipm(x2) - ipn(x2) ipm(x 1)] /V2,

respectively, are correctly symmetric and anti-symmetric. Fermions

The antisymmetric combination ipA) equation (5.5), is found to apply to electrons, and to other particles, including protons and neutrons, which are called fermions. By looking at ipA in the case m = n, one finds \pA= 0.

The wavefunction for two fermions in exactly the same state, is zero! This is a statement of the Pauli exclusion principle: only one Fermi particle can occupy a completely specified quantum state.

For other particles, notably photons of electromagnetic radiation, the symmetric combination ips(l,2), equation (5.4), is found to occur in nature. Macroscopically large numbers of photons can have exactly the same quantum state, and this is important in the functioning of lasers. Photons, alpha particles, and helium atoms are examples of Bose particles, or bosons.

The Bose condensation of rare gas atoms has been an important research area in the past several years. The superfluid state of liquid helium, and the superconducting ground state of electron pairs, can also be regarded as representing Bose condensates, in which macroscopically large numbers of particles have exactly the same quantum numbers.

Electron pairs and the Josephson effect are available only at low temperatures, but they enable a whole new technology of computation, in which heat generation (energy consumption) is almost zero. So the question, is a refrigerator worth its expense, is becoming more of a realistic economic choice, as energy density in the silicon technology continues to rise with Moore's Law.

Orbital and Spin Components of Wavefunctions

To return to covalent bonds of di-atoms, it is necessary to complete the description of the electron state, by adding its spin projection ms=± 1/2. It is useful to separate the space part <p(x) and the spin part x of the wavefunction, as

For a single electron x=t (for ms= 1/2) or x=^ (for = —1/2). For two electrons there are two categories, S = 1 (parallel spins) or S = 0 (antiparallel spins). While the S =0 case allows only ms = 0, the S = 1 case has three possibilities, ms= 1,0,-1, which are therefore referred to as constituting a "spin triplet".

A good notation for the spin state is Xs,m> so that the spin triplet states are

Bosons

Xi,i = 1YI* = and xi,0=1/^2 (T^ + ¿iT2) (spin triplet) (5.7)

For the singlet spin state, S = 0, one has

Inspection of these makes clear that the spin triplet (S = 1) is symmetric on exchange, and the spin singlet (S = 0) is antisymmetric on exchange of the two electrons.

Since the complete wavefunction (for a fermion like an electron) must be antisymmetric for exchange of the two electrons, this can be achieved in two separate ways:

ipA(\,l) = 0sym (1,2) Xanti (1,2) = 0sym (1,2) S = 0 (spin singlet) (5.9)

Va(1,2) = 0anti (1,2) Xsym (1,2) = 0anti (1,2) S = 1 (spin triplet). (5.10)

The structure of the orbital or space wavefunctions <psym,anti (1,2) here is identical to those shown above for 7ps>A(l,2), equations (5.4) and (5.5).

The Hydrogen Molecule, Di-hydrogen: The Covalent Bond

Consider two protons, (labeled a, and b, assume they are massive and fixed) a distance R apart, with two electrons. If R is large and we can neglect interaction between the two atoms, then, following the discussion of Tanner [1],

[-(h2/2m) (Vx2 + Va2) + U(ri) + U(r2)]ip= (Jf i+Jf2) V= (5-n)

where the rs represent the space coordinates of electrons 1 and 2. Solutions to this problem, with no interactions, can be ip=ipa(x1)ip\)(x2) or ip=i/>a(x2)ipb(xi) (with the wavefunction centered at proton a,b) and the energy in either case is E = Ea + E^.

The interaction between the two atoms is the main focus, of course. The interactions are basically of two types. First, the repulsive interaction ke2/r1>2r with r12 the spacing between the two electrons. Secondly, the attractive interactions of each electron with the 'second' proton, proportional to (l/ra>2 + 1/^b.i)- The latter attractive interactions, primarily occurring when the electron is in the region between the two protons, and can derive binding from both nuclear sites at once, stabilize the hydrogen molecule (but destabilize a ferromagnet).

Altogether we can write the interatom interaction as jeint= ke2[l/R +l/rlf2 - l/ra,2 - l/rb>1]. (5.12)

To get the expectation value of the interaction energy the integration, below, following equation (4.38) must extend over all six relevant position variables q. (Spin variables are not acted on by the interaction.)

The appropriate wavefunctions have to be overall antisymmetric. Thus, following equations (5.9) and (5.10), the symmetric 0sym(l,2) orbital for the antisymmetric S =0 (singlet) spin state, and the anti-symmetric 0antl(l,2) orbital for the symmetric S = 1 (triplet) spin state. The interaction energies are

<£int> = A2 (K1j2 + Jh2) for S = 0 (spin singlet) (5.14)

<£int> = B2 (Klf2 -J1>2) for S = 1 (spin triplet), where (5.15)

The physical system will choose, for each spacing R, the state providing the most negative of the two interaction energies. For the hydrogen molecule, the exchange integral Jlt2 is negative, so that the covalent bonding occurs when the spins are anti-parallel, in the spin singlet case.

Covalent bonding and covalent anti-bonding, purely nanophysical effects

The qualitative understanding of the result is that in the spin singlet case the orbital wavefunction is symmetric, allowing more electron charge to locate halfway between the protons, where their electrostatic energy is most favorable. A big change in electrostatic energy (about 2J1>2) is linked (through the exchange symmetry requirement) to the relative orientation of the magnetic moments of the two electrons. This effect can be summarized in what is called the exchange interaction

In the case of the hydrogen molecule, J is negative, giving a negative bonding interaction for antiparalld spins. The parallel spin configuration is repulsive or anti-honding. The difference in energy between the bonding and antbonding states is about 9 eV for the hydrogen molecule at its equilibrium spacing, R = 0.074 nm. The bonding energy is about 4.5 eV.

This is a huge effect, ostensibly a magnetic effect, but actually a combination of fundamental symmetry and electrostatics. The covalent bond is what makes much of matter stick together.

The covalent bond as described here is a short-range effect, because it is controlled by the overlap of the exponentially decaying wavefunctions from each nucleus. This is true even though the underlying Coulomb force is a long-range effect, proportional to 1/r2.

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