The quantized states of chemistry

What a nice illustration of the energy distribution in molecular systems! They even invented Boltzmann bucks, see their supplementary info )))

Density as a function of observables

For the state defined by the set of constraints = observables the mean values of these observables and their Lagrange multipliers are two conjugate variables. In order to define the first one - one needs the density, while for the second one - one needs the surprisal - the logarithm of the density:

Algebraic approach in molecules

For those of you who’s primary tool is not a computer but a pen and a paper: you may find these books very amusing (and maybe even useful):

C.E. Wullfman, Dynamical Symmetry (World Scientific Publishing, 2011)

W. Louisell, Quantum statistical properties of radiation (John Wiley & Sons, New York, 1973)

W. Miller, Jr., Lie Theory and Special Functions (Academic, New York, 1968).

Math is fun

Some relations in mathematics are so elegant! But only sometimes they are also useful : >

See some examples:

Faulhaber's Formula Gabriel's Staircase Nicomachus's Theorem and many more

Gaussian functions and their integrals

It is well-known that a normal distribution can be fully characterised by its first two moments. Its cumulants of the order higher than 2 are all zero (see for example, Abramowitz, Stegun).

However, in terms of the dynamics of the density which can be closely approximated by a Gaussian distribution it becomes an interesting feature. One can derive explicitly the relation between the mean values of the higher powers of coordinate and momentum using analytical form of the Gaussian integrals and their mean values and variance.

Relation between the Gaussian integrals needed to compute the mean values of different powers of R:

Harmonic vs. anharmonic motion

(Classical Dynamics) J. Chem. Phys. 101, 8768 (1994): In this paper M. Ben-Nun and R.D. Levine show one possible way to describe the harmonic motion of the Morse oscillator. One of the key steps was actually done years before, in 1978, in the paper by W.C. DeMarcus: Am. J. Phys. 46, 733 (1978).

(Quantum Dynamics) In the above mentioned paper it is shown that one can transform the Morse potential to the quadratic potential, however such a transformation will affect the commutation relations between the newly defined coordinate and momentum operators. The possible way to have closed algebra for the Morse potential was described by Wulfman and Kumei using the time-dilation technique. See the details in the recent book by Wulfman, Dynamical Symmetry (ch. 10.4) The derivation can be found in a Master thesis of Kumei.

Coherent states of the anharmonic oscillator were described earlier, by S. Kais and R.D. Levine, Phys. Rev. A 41, 2301. An interesting twist for the expression of these coherent states in energy domain can be found for the so-called directed states. See for example here: J. Phys. Chem. 1987, 91, 21, 5462-5465. It is important to remember that the number of bound states is finite, therefore infinite sum over the states will necessarily include the states of the dissociative continuum.