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)
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 a winner of the German Leibniz prize 2008 and the EPS Europhysics Prize 2014. He obtained the Leibniz prize among others for the achievement that, for the first time, he was able to create pictures of atoms with subatomic resolution. 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, not only physics.
The atomic shape that one can measure is actually a 3-dimensional charge density rho(x) (x in R^3).
''It is important to stress here that, unlike a many-electron wave function, the electron density is an observable; for example the X-ray scattering from an atomic or molecular gas gives rather direct information about the spatial distribution of electrons.'' (This is quoted verbatim from the introduction of the chapter N.H. March, Origins - The Thomas-Fermi theory, pp. 1-77 in: Theory of the Inhomogeneous Electron Gas. Physics of Solids and Liquids, Springer 1983.

In microscopic terms, the charge density rho(x) is formed by integrating the square of the absolute value of the 3n-dimensional wave function psi over 3n-3 dimensions.
The orbitals displayed 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.
More precisely, the charge density 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
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 the previous integral 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.

Viewing atoms or molecules with a scanning tunneling microscope (STM) or an atomic force microscope (AFM) 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 pictures ( more pictures). Thus rho(x) is actually observable, with a resolution of currently up to 0.6 Angstrom = 0.6 10^{-10}m. more pictures
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
or Bader's web site

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.

a discussion presenting a positive majority vote among 22 textbooks and from 2007.
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)
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 defends the unobservability of orbitals, but 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.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)

A theoretical physics FAQ