------------------------------------------
S4a. How do atoms and molecules look like?
------------------------------------------
Today, images of single atoms and molecules can be routinely produced.
M. Herz, F.J. Giessibl and J. Mannhart
Probing the shape of atoms in real space
Phys. Rev. B 68, 045301 (2003)
http://prola.aps.org/pdf/PRB/v68/i4/e045301
write in the introduction:
''quantum mechanics specifies the probability of finding an electron
at position x relative to the nucleus. This probability is
determined by |psi(x)|^2, where psi(x) is the wave function of the
electron given by Schroedinger's equation. The product of -e and
|psi(x)|^2 is usually interpreted as charge density, because the
electrons in an atom move so fast that the forces they exert on
other charges are essentially equal to the forces caused by a
static charge distribution -e|psi(x)|^2.''
One of the authors, Jochen Mannhart, is one of the 10 winners of the
Leibniz prize 2008,
http://www.dfg.de/aktuelles_presse/preise/leibniz_preis/2008/
among others for the achievement that, for the first time, he made
pictures of atoms with subatomic resolution possible.
The Leibniz prize is the highest German academic prize, endowed with
a research grant of up to 2.5 Million Euro for each winner,
awarded each year to a few excellent younger scientists from all
sciences.
The orbitals one can look at in physics and chemistry books
are the pictures of the squared absolute values of basis functions
used for representing single electron wave functions.
The actual shape of the wave function of each electron is some linear
combination of such basis function. These are calculated (in the
simplest realistic approximation) by Hartree-Fock calculations.
The atom shape is the shape of all electrons together, forming
in the Hartree-Fock approximation a Slater determinant formed from the
single-particle wave functions, and in general a linear combinations
of such Slater determinants. These live in a multidimensional space
with 3n dimensions for an atom with n electrons.
The shape one can measure is actually a 3-dimensional charge density
rho(x) (x in R^3) formed by integrating the square of the absolute
value of the 3n-dimensional wave function psi over 3n-3 dimensions.
[See, e.g., Slide 19 of
http://fds.oup.com/www.oup.com/pdf/acad/Lectures_Slides.pdf ]
More precisely, rho(x) is defined (nonrelativistically) such that
(apart form a constant factor and the charge contribution of the
nucleus)
integral dx rho(x) f(x) = psi^* O_1(f) psi (1)
for all nice 3-dimensional functions f(x) of the space coordinate
vector x, where
O_1(f) = integral f(x) a^*(x) a(x)
is the 1-particle operator corresopnding to f. Here a^* and a denote
creation and annihilation operators. Since rho(x) decays quickly as x
differs more and more from the atom center, the atom looks like a
charge cloud with slightly fuzzy boundary.
For isolated atoms in the absence of external fields,
rho is typically spherically symmetric, giving symmetric shapes.
(In case of particles of nonzero spin, this assumes
that we are in a thermal setting where the spin directions average out.
In this case, we have instead of (1) the formula
integral dx rho(x) f(x) = tr O_1(f) rhohat,
where rhohat is the density matrix of the mixed state.)
For molecules, rho is in fact also a function of the coordinates of
all nuclei involved, and there is no longer any reason to have more
symmetry than the symmetry of the configuration of nuclei,
which is very little and often none.
The shape of molecules is therefore mainly determined by the geometry
of the positions of the nuclei. In equilibrium, these arrange
themselves such that the potential energy, i.e., the smallest
eigenvalue of the Hamiltonian operator for the electrons is minimal
among all other positions (or at least a local minimum from which a
deeper lying state is very difficult to reach). The charge density
of molecules can be identified by means of X-ray crystallography or
nuclear magnetic resonance (NMR) spectroscopy; however, for complex
molecules, doing this reliably from the available indirect information
is a highly nontrivial art.
A few years ago,
I wrote a survey of molecular modeling of proteins, the largest
molecules in nature (apart from crystals, which are essentially
molecules of macroscopic size):
A. Neumaier,
Molecular modeling of proteins and mathematical prediction of
protein structure,
SIAM Review 39 (1997), 407-460.
http://www.mat.univie.ac.at/~neum/papers/physpapers.html#protein
Viewing atoms or molecules with a scanning tunneling microscope (STM)
or an atomic force microscope (AFM)
http://en.wikipedia.org/wiki/Atomic_force_microscope
amounts to scanning the response of the 3-dimensional charge density
to (or, more precisely, the current or force induced by it on)
the scanning device, from which a computer generates a picture.
http://www.physics.purdue.edu/nanophys/images.html
http://www.almaden.ibm.com/vis/stm/gallery.html
Thus rho(x) is actually observable, with a resolution of currently
up to 0.6 Angstrom = 0.6 10^{-10}m.
http://www.hypography.com/article.cfm?id=34288
For a discussion of the charge density of molecules and the resulting
operative interpretation of atoms in molecules see, e.g., the
encyclopedic article
R.F.W. Bader
Atoms in Molecules
http://pubs.acs.org/doi/abs/10.1021/ar00109a003
http://59.77.33.35/non-cgi/usrd8wqiernb/5/20/Atoms20in20Molec_1193580192.pdf
or Bader's web site
http://www.chemistry.mcmaster.ca/faculty/bader/aim/aim_0.html
On the other hand, whether atomic or molecular substructures such as
orbitals are observable is controversial. See, e.g.,
J.M. Zuo et al.,
Direct Observation of d-orbital holes and Cu-Cu bonding in Cu_2O,
Nature 401 (1999), 49-52.
http://www.nature.com/nature/journal/v401/n6748/pdf/401049a0.pdf
http://www.nature.com/nature/journal/v401/n6748/pdf/401021a0.pdf
http://www.public.asu.edu/~jspence/NewsViews.pdf
http://web.missouri.edu/~glaserr/412f99/synopsis_nv1.pdf
http://philsci-archive.pitt.edu/archive/00000228/00/Orbital_Observed.pdf
https://jchemed.chem.wisc.edu/HS/Journal/Issues/2001/Jul/abs877_2.html
http://philsci-archive.pitt.edu/archive/00001077/00/Jenkins.doc
for discussions in 1999-2001, a discussion presenting a positive
majority vote among 22 textbooks:
http://wwwcsi.unian.it/educa/inglese/halfacen.html
and from 2007:
http://jjap.ipap.jp/link?JJAP/46/L161/
Also, see the nice pictures in
M. Herz, F.J. Giessibl and J. Mannhart
Probing the shape of atoms in real space
Phys. Rev. B 68, 045301 (2003)
http://prola.aps.org/pdf/PRB/v68/i4/e045301
Apparently, it is a matter of terminology. Those who use the term
orbital to refer to a charge distribution corresponding to a particular
electronic state (and the ball- dumbbell-, or ring-shaped pictures of
orbitals in textbooks show just that) find orbitals observable, while
purists restricting the usage of orbitals to denoting particular
single-electron wave functions find them unobservable.
Note that Scerri, who in
http://philsci-archive.pitt.edu/archive/00000228/00/Orbital_Observed.pdf
defends the unobservability of orbitals, writes explicitly:
''What can be observed, and frequently is observed in experiments, is
electron density. In fact, the observation of electron density is a
major field of research in which several monographs and review
articles have been written.''
and then cites two books and a review article. A more recent review
article of some aspects is
J.M. Zuo
Measurements of electron densities in solids: a real-space view of
electronic structure and bonding in inorganic crystals
Rep. Progr. Phys. 67 (2004), 2053-2103.