# Do different vibrational modes correspond to vibrational energy levels?

Do different vibrational modes correspond to vibrational energy levels? Or all modes can be found in ground state of the molecule? Or each vibration for each bond has its own vibrational energy levels? I don't understand, please help.

• As a generality different vibrational modes will have different energies. However it is possible that two different modes have the same energy. // In the ground state there would be no molecular vibrations. // Sometimes the molecular vibration can be considered as only one bond vibrating. // Look at Infrared Spectroscopy for more details en.wikipedia.org/wiki/Infrared_spectroscopy – MaxW Jan 31 '17 at 22:29
• @MaxW Are you claiming that "in the ground state there would be no molecular vibrations"? – Zhe Jan 31 '17 at 23:20
• There are molecular vibrations in the ground state. I've searched a bit and apparently each vibrational mode has its own ground frequency and energy and different vibrational energy levels. – NeneS Jan 31 '17 at 23:44
• Each vibrational mode has its characteristic frequency $\nu$ (usually reported as the wavenumber $\bar{\nu} = \nu/c$, $c$ being the speed of light). That's all, really; the rest follows from this, since the ground-state energy (zero-point energy) is $h\nu/2$, the vibrational spacings are $h\nu$ and the energy levels are $(n+\frac{1}{2})h\nu$ where $h$ is Planck's constant and $n$ the vibrational quantum number. – orthocresol Jan 31 '17 at 23:53
• @MaxW there exists a zero-point energy (as an example, consider the quantum harmonic oscillator) – GnomeSort Feb 1 '17 at 7:15

yes and no: all vibrational normal modes have a different energy (with the proviso that some may be accidentally degenerate) and hence different absorption frequency. (Fundamental frequencies have vibrational transitions that occur with selection rule $\Delta n=\pm 1$.) All normal modes exist in their ground state with zero point energy (quantum number $n=0$) as well as in other levels based on the same mode ($n>0$) which are populated according the Boltzmann distribution at temperature T. Thus each basic vibrational mode i has its own stack of energy levels, with quantum numbers $n= 0, 1, 2 \cdot\cdot$ and (harmonic oscillator) energy $E_i=\nu_i(n_i+1/2)$ with quantum numbers $n_i$ and frequency $\nu_i$ in $\pu {cm^{-1}}$. The total zero point energy of the molecule is the sum of all the individual zero point energies above the zero of energy.

The number of vibrational levels populated at a given temperature depends on the vibrational level spacing, thus low frequency modes, say, $500 \pu{cm^{-1}}$ have far more levels populated than a high frequency, say $1500 \pu{cm^{-1}}$ mode.

As the energy increases the number of vibrational energy levels / unit energy can become very large indeed in polyatomic molecules. There are overtones where only one vibrational mode is excited, but with more than one quantum of energy, and combinations where two or more different vibrations of different frequency are excited and so each have non zero quantum numbers.

In water molecules there are 3 vibrations (of frequency approx $3650,1595, 3756 \pu{cm^{-1}}$) if we label them in order with quantum numbers then [000] is the zero point energy $(3650+1595+3756)/2=4500 \pu{cm^{-1}}$, [010] a fundamental in the second frequency and [020] an overtone in this mode and [110] a combination of modes 1 and 2 with energy above zero of $4500+3650+1595 \pu{cm^{-1}}$

Answer to your question. Yes and no. Different vibrational modes may correspond to different energies and may not (it all depends on the molecule and the vibrations concerned). The energies may come to be very close sometimes (degeneracy).

Explanation:

Any molecule has certain number of vibrations possible. These are given by, $$(3N-5)$$ for linear molecules and $$(3N-6)$$ for non-linear molecules. These vibrations are called vibrational modes and correspond to distinct types of vibrations (careful, we have not discussed energies yet).

As a simple approximation, the vibration can be modeled as harmonic oscillator. Then each of the mode is an individual harmonic oscillator. (In a real molecules these are not entirely independent but we are working under approximation). Then, if we consider energies corresponding to the vibrational states (regardless of the mode it comes from) in increasing order, we get a table of energies (which are observables either in IR or Raman spectra or in both IR and Raman). In literature these energies are expressed in units of wavenumbers ($$cm^{-1}$$).

• In experimental vibrational spectroscopy, we measure energy differences between the initial and final state and hence we have measurement of energy of first state, second state and so on. The ground state's energy is measured by other techniques. Due to zero point energy it is non zero and has some value in appropriate energy units.

To be more clear, I give example:

i) water: The following data is taken from this paper (http://www.sciencedirect.com/science/article/pii/S0022407312004311) from page 45.

From the above image you can see that, the energy levels are arranged in increasing order. $$\nu_{1}\nu_{2}\nu_{3}$$ are the vibrational modes (or individual oscillators), and the states occupied are mentioned using numbers (000), (010) and so on. ($$\nu_{1}$$=symmetric stretch; $$\nu_{2}$$=anti-symmetric stretch ; $$\nu_{3}$$=bending).