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If we consider N atoms in a non-linear molecule, then each atom can move independently on the X, Y, and Z axes. That's $3D \times N = 3N$ degrees of freedom.

But if I move the whole molecule along the x-axis $3 \overset{\lower.5em\circ}{\mathrm{A}}$ that doesn't constitute a vibration - it's just a translation. So we have to remove 3 degrees of freedom for translations.

Similarly, if I rotate the molecule along the x- or y- or z-axis, that's not a vibration either. You mentioned water, which is non-linear, so I remove another 3 degrees of freedom for rotations.

I'm left with $3N-6$ degrees of freedom - which are the normal modes of vibration. Your question then becomes "what are those normal modes" (e.g., for water).

Okay, for water, we have $9 - 6 = 3$ normal modes. As mentioned above in a comment, it's not like you can just pull an atom along an x-axis.. As it turns out, the normal modes will be dictated by the potential energy surfaces of the molecule.

It's been a while since I've taught this, but in general, one normal mode is a "symmetric stretch" - usually expanding and contracting bonds. Other modes will depend on the molecule, but will depend on the geometry/bonding and elements involved: stretching (symmetric and asymmetric), bending (e.g. changing the bond angle in water), out-of-plane bending (e.g. flexing a benzene ring), etc.

Note: The reason I mention normal modes in regards to the question about "normal coordinates" is that the two are directly linked. The "normal coordinates" are the displacements along the normal modes in a molecular vibration:

The normal coordinates, denoted as ''Q'', refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time.

The reason that we use the term is that the Cartesian x, y, and z axes are meaningless with respect to a molecular geometry, whereas the normal modes and thus the normal coordinates are derived directly from the molecular potential energy surface.

If we consider N atoms in a non-linear molecule, then each atom can move independently on the X, Y, and Z axes. That's $3D \times N = 3N$ degrees of freedom.

But if I move the whole molecule 3 Å along the x-axis, that doesn't constitute a vibration - it's just a translation. So we have to remove 3 degrees of freedom for translations.

Similarly, if I rotate the molecule along the x- or y- or z-axis, that's not a vibration either. You mentioned water, which is non-linear, so I remove another 3 degrees of freedom for rotations.

I'm left with $3N-6$ degrees of freedom - which are the normal modes of vibration. Your question then becomes "what are those normal modes" (e.g., for water).

Okay, for water, we have $9 - 6 = 3$ normal modes. As mentioned above in a comment, it's not like you can just pull an atom along an x-axis.. As it turns out, the normal modes will be dictated by the potential energy surfaces of the molecule.

In general, one normal mode is a "symmetric stretch" - usually expanding and contracting bonds. Other modes will depend on the molecule, but will depend on the geometry/bonding and elements involved: stretching (symmetric and asymmetric), bending (e.g. changing the bond angle in water), out-of-plane bending (e.g. flexing a benzene ring), etc.

Note: The reason I mention normal modes in regards to the question about "normal coordinates" is that the two are directly linked. The "normal coordinates" are the displacements along the normal modes in a molecular vibration:

The normal coordinates, denoted as ''Q'', refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time.

The reason that we use the term is that the Cartesian x, y, and z axes are meaningless with respect to a molecular geometry, whereas the normal modes and thus the normal coordinates are derived directly from the molecular potential energy surface.

If we consider N atoms in a non-linear molecule, then each atom can move independently on the X, Y, and Z axes. That's $3D \times N = 3N$ degrees of freedom.

But if I move the whole molecule along the x-axis $3 \overset{\lower.5em\circ}{\mathrm{A}}$ that doesn't constitute a vibration - it's just a translation. So we have to remove 3 degrees of freedom for translations.

Similarly, if I rotate the molecule along the x- or y- or z-axis, that's not a vibration either. You mentioned water, which is non-linear, so I remove another 3 degrees of freedom for rotations.

I'm left with $3N-6$ degrees of freedom - which are the normal modes of vibration. Your question then becomes "what are those normal modes" (e.g., for water).

Okay, for water, we have $9 - 6 = 3$ normal modes. As mentioned above in a comment, it's not like you can just pull an atom along an x-axis.. As it turns out, the normal modes will be dictated by the potential energy surfaces of the molecule.

It's been a while since I've taught this, but in general, one normal mode is a "symmetric stretch" - usually expanding and contracting bonds. Other modes will depend on the molecule, but will depend on the geometry/bonding and elements involved: stretching (symmetric and asymmetric), bending (e.g. changing the bond angle in water), out-of-plane bending (e.g. flexing a benzene ring), etc.

If we consider N atoms in a non-linear molecule, then each atom can move independently on the X, Y, and Z axes. That's $3D \times N = 3N$ degrees of freedom.

But if I move the whole molecule along the x-axis $3 \overset{\lower.5em\circ}{\mathrm{A}}$ that doesn't constitute a vibration - it's just a translation. So we have to remove 3 degrees of freedom for translations.

Similarly, if I rotate the molecule along the x- or y- or z-axis, that's not a vibration either. You mentioned water, which is non-linear, so I remove another 3 degrees of freedom for rotations.

I'm left with $3N-6$ degrees of freedom - which are the normal modes of vibration. Your question then becomes "what are those normal modes" (e.g., for water).

Okay, for water, we have $9 - 6 = 3$ normal modes. As mentioned above in a comment, it's not like you can just pull an atom along an x-axis.. As it turns out, the normal modes will be dictated by the potential energy surfaces of the molecule.

It's been a while since I've taught this, but in general, one normal mode is a "symmetric stretch" - usually expanding and contracting bonds. Other modes will depend on the molecule, but will depend on the geometry/bonding and elements involved: stretching (symmetric and asymmetric), bending (e.g. changing the bond angle in water), out-of-plane bending (e.g. flexing a benzene ring), etc.

Note: The reason I mention normal modes in regards to the question about "normal coordinates" is that the two are directly linked. The "normal coordinates" are the displacements along the normal modes in a molecular vibration:

The normal coordinates, denoted as ''Q'', refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time.

The reason that we use the term is that the Cartesian x, y, and z axes are meaningless with respect to a molecular geometry, whereas the normal modes and thus the normal coordinates are derived directly from the molecular potential energy surface.

If we consider N atoms in a non-linear molecule, then each atom can move independently on the X, Y, and Z axes. That's $3D \times N = 3N$.

But if I move the whole molecule along the x-axis $3 \AA$ that doesn't constitute a vibration - it's just a translation. So we have to remove 3 degrees of freedom for translations.

Similarly, if I rotate the molecule along the x- or y- or z-axis, that's not a vibration either. You mentioned water, which is non-linear, so I remove another 3 degrees of freedom for rotations.

I'm left with $3N-6$ degrees of freedom - which are the normal modes of vibration. Your question then becomes "what are those normal modes" (e.g., for water).

Okay, for water, we have $9 - 6 = 3$ normal modes. As mentioned above in a comment, it's not like you can just pull an atom along an x-axis.. As it turns out, the normal modes will be dictated by the potential energy surfaces of the molecule.

It's been a while since I've taught this, but in general, one normal mode is a "symmetric stretch" - usually expanding and contracting bonds. Other modes will depend on the molecule, but will depend on the geometry/bonding and elements involved: stretching (symmetric and asymmetric), bending (e.g., changing the bond angle in water), out-of-plane bending (e.g., flexing a benzene ring), etc.

If we consider N atoms in a non-linear molecule, then each atom can move independently on the X, Y, and Z axes. That's $3D \times N = 3N$ degrees of freedom.

But if I move the whole molecule along the x-axis $3 \overset{\lower.5em\circ}{\mathrm{A}}$ that doesn't constitute a vibration - it's just a translation. So we have to remove 3 degrees of freedom for translations.

Similarly, if I rotate the molecule along the x- or y- or z-axis, that's not a vibration either. You mentioned water, which is non-linear, so I remove another 3 degrees of freedom for rotations.

I'm left with $3N-6$ degrees of freedom - which are the normal modes of vibration. Your question then becomes "what are those normal modes" (e.g., for water).

Okay, for water, we have $9 - 6 = 3$ normal modes. As mentioned above in a comment, it's not like you can just pull an atom along an x-axis.. As it turns out, the normal modes will be dictated by the potential energy surfaces of the molecule.

It's been a while since I've taught this, but in general, one normal mode is a "symmetric stretch" - usually expanding and contracting bonds. Other modes will depend on the molecule, but will depend on the geometry/bonding and elements involved: stretching (symmetric and asymmetric), bending (e.g. changing the bond angle in water), out-of-plane bending (e.g. flexing a benzene ring), etc.