# Can one compute the vibrational spectrum of a bond by the Fourier transform of the dipole moment vector autocorrelation function $C_{\mu\mu}(t)$?

Is it true that one can calculate the vibrational spectrum of a bond by the Fourier transform of the dipole moment vector autocorrelation function $C_{\mu \mu}(t)$?

For example, suppose that I have a diatomic molecule $\text{A-B}$, containing atoms $\text{A}$ and $\text{B}$.

$\text{A}$ and $\text{B}$ have partial charges $q_A$ and $q_B$ that sum to zero: $q_A + q_B = 0$.

I can find the dipole moment vector $\vec{\mu}$ of this diatomic molecule:

$$\vec{\mu}(t) = \sum_i q_i \vec{r}_i^{\prime} = q_A \vec{r}_A^{\prime} + q_B \vec{r}_B^{\prime}$$

where $\vec{r}_A^{\prime}(t)$ and $\vec{r}_B^{\prime}(t)$ are the position vectors of the charges (atoms). These position vectors depend on time $t$ because the charges (atoms) move -- the bond vibrates. Thus the dipole moment vector $\vec{\mu}(t)$ also depends on time $t$.

I think that the autocorrelation function $C_{\mu\mu}(t)$ can be written

$$C_{\mu\mu}(t) = \langle \vec{\mu}(t) \cdot \vec{\mu}(0) \rangle$$

where $\langle ... \rangle$ denotes an ensemble average.

Now, $C_{\mu\mu}(t)$ is in the time domain. It has dimensions of $\text{(dipole moment)}^2$ (typically, $\text{Debye}^2$ in chemistry). But I would like to compute the vibrational spectrum (a plot of intensity, arbitrary units, etc. versus frequency). Is it true that the vibrational spectrum is the Fourier transform of $C_{\mu\mu}(t)$?

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It is the imaginary part of the Fourier transform. – Fabian Dec 15 '12 at 20:01
This isn't right. – KDN Dec 16 '12 at 4:20

The power-spectral density is the Fourier transform of the autocorrelation function. It is not simply the imaginary part; it will depend on your data. Your data is real, so the spectrum is hermitian, i.e., $A(z)=A(-z)*$. If your data is an even function (symmetric about 0, like cosine), then the transform will be purely real. If it is an odd function, (like sine) then the transform will be purely imaginary. If it is neither even nor odd, then the transform will have both real and imaginary parts.
When you want to know what the power spectrum for a periodic signal is, you generally mean to include the power at both the frequency and it's negation, i.e., colloquially, the "power at frequency $f$" is actually $P(f)+P(-f)$. It is this condition that keeps you from having imaginary power, since your real-valued data has a hermitian Fourier transform.