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In summary, if you know the energy and bond length of a covalent bond, you can find the harmonic-oscillator frequencies by using the second-order derivatives of energy with respect to bond length.

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You can't.

If you know the energy as a function of bond length E(r), then you can determine the harmonic-oscillator eigenfrequencies from the second-order derivatives of E with respect to r, taken at the equilibrium point, where the first-order derivatives (forces) are zero. Getting anharmonic corrections requires even higher-order derivatives.

Or, to put it another way, the vibrational frequencies depend on (and are in fact a kind of description of) the potential-energy curve E(r) of the bond distance. You need to know the whole curve to know the frequencies. If you only know the energy and bond length, all you have is a single point on the curve. The simplest way to get the information about it is to determine the derivatives as above, in what's essentially a Taylor expansion around the equilibrium point.

You can't get E(r) with anything less than a full quantum-mechanical calculation.

If you know the energy as a function of bond length E(r), then you can determine the harmonic-oscillator eigenfrequencies from the second-order derivatives of E with respect to r, taken at the equilibrium point, where the first-order derivatives (forces) are zero. Getting anharmonic corrections requires even higher-order derivatives.

Or, to put it another way, the vibrational frequencies depend on (and are in fact a kind of description of) the potential-energy curve E(r) of the bond distance. You need to know the whole curve to know the frequencies. If you only know the energy and bond length, all you have is a single point on the curve. The simplest way to get the information about it is to determine the derivatives as above, in what's essentially a Taylor expansion around the equilibrium point.

You can't get E(r) with anything less than a full quantum-mechanical calculation.

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The best you could do is fit an idealized potential.

For example, if you think about the Lennard-Jones potential. There are two parameter that are fit from experiment. If you have the position and energy of the minima then you have enough to make the potential fit.

If you knew that it was s-valent you could fit the overlap integral for a LCAO model. In is textbook David Pettifor suggests that you can use a power series to describe the integral's dependence on r. Walter Harrison also had some useful function forms in his book, I think. Of course you don't have enough information to fit anything with p or d orbitals.

Of course both of these approximations are severe and wouldn't stand in a research setting.

For example, if you think about the Lennard-Jones potential. There are two parameter that are fit from experiment. If you have the position and energy of the minima then you have enough to make the potential fit.

If you knew that it was s-valent you could fit the overlap integral for a LCAO model. In is textbook David Pettifor suggests that you can use a power series to describe the integral's dependence on r. Walter Harrison also had some useful function forms in his book, I think. Of course you don't have enough information to fit anything with p or d orbitals.

Of course both of these approximations are severe and wouldn't stand in a research setting.

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