The zero-point vibrational energy (ZPVE) is the non-zero amount of non-electronic energy that molecules have even at zero kelvin and is purely quantum mechanical in nature. For a molecule with $c$ countable normal modes, it is defined as

$$ E_{\text{ZPVE}} = \frac{1}{2} \hbar \sum_{c} \omega_{c} $$

where $\omega_{c}$ is the energy or frequency of each normal mode.

Intensive and extensive variables have always confused me. This reference gives the following concise bulleted definition:

  • Extensive:
    • Additive
    • State variables (only sometimes)
  • Intensive:
    • Not additive
    • Field variables
    • Point variables (i.e., same at each point in a system at equilibrium)

with further examples of each (also on Wikipedia):

  • Extensive: $Q$, $W$, $U$ (and other free energies), $N$, $V$, $M$, $C$, $n$
  • Intensive: $P$, $T$, $\mu$, $\xi$ (electric field), any field variable, ratio of extensive/extensive

To use an example, the total energy of a system is extensive. If I have one molecule of caffeine, calculate $U$, and then take two caffeine molecules (even at infinite separation), and calculate $U$, they'll be different.

Now consider the ZPVE. For a nonlinear molecular system, there are $3N-6$ normal modes, where $N$ is the number of atoms. Take caffeine again, which has 24 atoms. There are 66 normal modes for one molecule and 138 for two molecules. If the ZPVE is considered to be a bulk or macroscopic property, then clearly it increases as the total number of normal modes increases, independent of whether or not the resulting normal modes are localized to one molecule. If the ZPVE is considered to be a single-molecule property (I hesitate to call it microscopic), then a single molecule is a single molecule, and it would make sense to "isolate" it in a way that doesn't make sense for other properties; that is, it doesn't make sense to only consider the temperature of a single molecule assumed to be in isolation that would in fact be part of some bulk phase.

So, perhaps a better wording of the question is: is ZPVE a microscopic (ugh) or macroscopic quantity? A property like polarizability also exists in a weird gray area for me as well.


2 Answers 2


Mathematically, an extensive property $f$ is one for which $f(\lambda x,\lambda y,z) = \lambda f(x,y,z)$ for extensive variables $x$, $y$, and intensive variable $z$. For example, the Helmholtz free energy $A(T,\lambda V,\lambda N) = \lambda A(T,V,N)$ is extensive.

Following this definition, the zero-point energy $$E_\text{ZPVE}(\{\omega_c\},\lambda n) = \lambda E_\text{ZPVE}(\{\omega_c\},n),$$ where $n$ is the number of normal modes, is extensive in the thermodynamic limit $N \to \infty$.

The polarizability should be able to be treated similarly.

It bears noting that a lot of macroscopic extensive properties are not extensive in the microscopic regime. For example, if we consider two identical systems, each with an energy $U$, then the combined system has energy $2U$ plus an additional energetic term $U_\text{sys-sys}$, accounting for superficial energetic interactions between the two subsystems. The surface, and hence the superficial energy term, scales as one dimension removed of the volume, and likewise vanishes in the thermodynamic limit.

Furthermore, it's perfectly valid to have properties that are neither extensive nor intensive, like the number of microstates that a system possesses.

  • $\begingroup$ ...could I have read all this in Ken Dill's book? $\endgroup$ Commented Jun 16, 2017 at 22:29
  • $\begingroup$ @pentavalentcarbon, maybe, I don't have that book. My standard reference is Chandler. $\endgroup$ Commented Jun 17, 2017 at 2:15

Is the zpe not a term in U just like other vibrational and rotational energy levels? The word 'internal' is confusing in molecular terms as it refers to a thermodynamic 'system' not molecules (which do not need to exist as far as thermodynamics goes). The internal energy is defined by First Law and so for an ideal/perfect gas of point like atoms it can have no other energy than from translational motion. After it was discovered that molecule have vibrational/rotational energy then it is possible that these energies can be changed by affecting the heat absorbed by and work done on the 'system' and so are part of the system's internal energy.

The point about intensive variables is that their value does not depend on how much 'stuff' there is, e.g. temperature and pressure.


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