Molecular vibrations and rotations
Prof. dr. Ad van der Avoird
January 28, 2010 Contents
0.1 Books ...... 2 0.2 Introduction...... 2 0.3 Vibrations of polyatomic molecules (harmonic) ...... 3 0.3.1 Harmonic oscillator in one dimension x ...... 3 0.3.2 Twocoupledoscillators...... 5 0.3.3 Harmonic vibrations of molecules in mass-weighted co- ordinates...... 8 0.3.4 Harmonic vibrations of molecules in cartesian (displace- ment)coordinates...... 12 0.3.5 Invariance conditions on the matrix F ...... 14 0.3.6 Vibrations in internal coordinates (3n-6) ...... 18 0.4 Rotations of rigid (non-linear) molecules ...... 21 0.4.1 Rotation matrices and infinitesimal rotations ...... 22 0.4.2 The rigid rotor in classical mechanics ...... 24 0.4.3 TherigidrotorHamiltonian ...... 31 0.4.4 The rigid rotor in quantum mechanics ...... 34 0.5 The molecular vibration-rotation Hamiltonian ...... 38 0.5.1 Introduction...... 38 0.5.2 The (classical) kinetic energy of a semi-rigid molecule . 39 0.5.3 Semi-rigid molecules in quantum mechanics ...... 44 0.5.4 Floppy molecules and Van der Waals molecules . . . . 46
1 0.1 Books
1. Molecular vibrations, E.B. Wilson, J.C. Decius, and P.C. Cross, McGraw- Hill, New York (1955)
2. Vibrational states, S. Califano, Wiley, London (1976)
3. Molecular Vibrational-Rotational spectra, D. Papousek and M.R. Aliev, Elsevier, Amsterdam (1982)
4. (Angular momentum in quantum physics, L.C. Biedenharn and J.D. Louck, Addison-Wesley, Reading (1981))
0.2 Introduction
Nuclear motion — Born-Oppenheimer approximation, second step
For n nuclei with coordinates rk,k = 1,...,n the Schr¨odinger equation for the nuclear motion problem is given by: ˆ H(r1,...,rn)Ψ(r1,...,rn)= EΨ(r1,...,rn) (1) with the Hamiltonian Hˆ consisting of the kinetic and potential energy oper- ators: ˆ ˆ ˆ H(r1,...,rn)= T(r1,...,rn)+ V(r1,...,rn). (2)
The potential energy V (r1,...,rn) can be obtained from the electronic Schr¨odinger equation (Born-Oppenheimer approximation, first step). It is an energy eigenvalue of the electronic Hamiltonian with coordinates of the nuclei fixed at rk,k =1,...,n. Variation of the nuclear coordinates rk yields the poten- tial energy surface V (r1,...,rn). Solution of the nuclear motion Schr¨odinger equation for molecules translations, rotations, vibrations →
Spectroscopy: microwave, submilimeter, IR / Raman
0 30 cm−1 rotations − 100 5000 cm−1 vibrations −
2 Lasers very high resolution (in principle) (7 significant→ digits)
Doppler and collision broadening occurring in the gas phase are avoided in molecular beams (“supersonic nozzle beams”: very low T)
Globally:
rotational spectrum molecular structure → vibrational spectrum intramolecular forces (“force field”) associated with chemical bonds → spectra of van der Waals molecules intermolecular (noncovalent) → forces
0.3 Vibrations of polyatomic molecules (harmonic)
0.3.1 Harmonic oscillator in one dimension x
Momentum operator, in atomic units (¯h = 1):
1 d ˆp = (3) x i dx The Hamiltonian of a harmonic oscillator is: ˆp2 1 1 d2 1 H=ˆ T+ˆ V=ˆ x + f ˆx2 = + f ˆx2, (4) 2m 2 −2m dx2 2 d2V dV where f = is the force constant. It is assumed that = 0 (potential dx2 dx minimum) and that higher than second derivatives of V can be neglected. It is convenient to introduce mass-weighted coordinates:
q = √mx (5) 1 d 1 d ˆp= = (6) i√m dx i dq
3 and write: 1 1 H=ˆ ˆp2 + λ ˆq2 (7) 2 2 f where λ = m The frequency of the harmonic oscillator is ω0 = √λ. 1 Eigenvalues are the energies E = ω (v + ) v =0, 1, 2,... (8) v 0 2 Eigenfunctions are Ψ (q)= H (√ω q) exp 1 ω q2 , (9) v v 0 − 2 0 where Hv are Hermite polynomials.
4 0.3.2 Two coupled oscillators
Example 1:
Figure 1: Linear CO2 with two internal coordinates.
Example 2:
Figure 2: Diatomic molecule in laboratory coordinates.
2 2 ˆ ˆp1 ˆp2 1 2 1 2 H= + + f11 ˆq1 + f22 ˆq2 +f12 ˆq1 ˆq2 (10) 2m1 2m2 2 2
Let us assume symmetry, so that m1 = m2 = m and f11 = f22. Define new coordinates:
Q1 =(q1 + q2)/√2 q1 =(Q1 + Q2)/√2 (11) Q =(q q )/√2 q =(Q Q )/√2 (12) 2 1 − 2 2 1 − 2 The associated momentum operators are: 1 ∂ 1 ∂q ∂ ∂q ∂ Pˆ = = 1 + 2 = (ˆp + ˆp )/√2 (13) 1 i ∂Q i ∂Q ∂q ∂Q ∂q 1 2 1 1 1 1 2 ˆ 1 ∂ P2 = = = (ˆp1 ˆp2)/√2 (14) i ∂Q2 −
5 Hence:
1 ∂ 1 ∂Q1 ∂ ∂Q2 ∂ ˆ ˆ ˆp1 = = + = P1 + P2 /√2 (15) i ∂q1 i ∂q1 ∂Q1 ∂q1 ∂Q2 1 ∂ ˆ ˆ ˆp2 = = = P1 P2 /√2 (16) i ∂q2 − Substitution of these new coordinates gives:
1 2 2 1 2 H=ˆ Pˆ + Pˆ + Pˆ Pˆ + f Qˆ + Qˆ + 4m 1 2 1 − 2 4 11 1 2 (17) 1 1 2 + f Qˆ + Qˆ Qˆ Qˆ + f Qˆ Qˆ 2 12 1 2 1 − 2 4 22 1 − 2 1 ˆ 2 ˆ 2 1 ˆ 2 = P1 + P2 + (f11 + f22 +2f12) Q1 + 2m 4 (18) 1 2 1 + (f + f 2f ) Qˆ + (f f ) Qˆ Qˆ 4 11 22 − 12 2 2 11 − 22 1 2
The last term vanishes because f11 = f22 and we get:
1 2 1 2 1 2 1 2 H=ˆ Pˆ + (f + f ) Qˆ + Pˆ + (f f ) Qˆ = Hˆ + Hˆ . (19) 2m 1 2 11 12 1 2m 2 2 11 − 12 2 1 2 As one can see, our system separates into two uncoupled oscillators, with:
coordinates Q1 =(q1 + q2)/√2 Q2 =(q1 q2)/√2 (20) 1 − 1 f + f /2 f f /2 frequencies ω = 11 12 ω = 11 − 12 (21) 1 m 2 m The energy of the system is:
1 1 E = Ev1 + Ev2 = ω1 v1 + 2 + ω2 v2 + 2 (22)
6 Example 1: CO2
Figure 3: Q1 and Q2 for CO2.
Q1 =(q1 + q2)/√2 — symmetric stretch vibration Q =(q q )/√2 — asymmetric stretch vibration 2 1 − 2
Q1 and Q2 are called “normal mode” coordinates
Example 2: Diatomic molecule
Figure 4: Q1 and Q2 for a diatomic molecule.
Q1 =(q1 + q2)/√2 — overall translation (center of mass motion) Q =(q q )/√2 — stretch vibration (relative motion) 2 1 − 2
In Example 2 the molecule (center of mass) freely moves in space, the en- ergy does not depend on Q1, and we get ω1 = 0. This corresponds to the invariance condition:
f11 + f12 (= f12 + f22) =0. (23)
7 For the stretch vibration Q =(q q )/√2 we find: 2 1 − 2 1 1 1 f f /2 2f /2 f /2 ω = 11 − 12 = 11 = 11 , (24) 2 m m µ m where µ = is the reduced mass. 2 Conclusion: Both examples show that the simple transformation Q = (q q )/√2 1,2 1 ± 2 separates the potential energy, while the kinetic energy remains separated. As a result we get two uncoupled harmonic oscillators.
0.3.3 Harmonic vibrations of molecules in mass-weighted coordinates
Coordinates of atoms:
rk = ak + dk (k =1,...,n) (25) where ak denotes equilibrium positions and dk stands for displacements. The components of the coordinates are:
xk akx dkx y = a + d (26) k ky ky zk akz dkz Altogether:
rkα = akα + dkα k =1,...,n; α =1, 2, 3(x, y, z) (27)
ri = ai + di i =1,...... , 3n (28)
Atomic masses: mk, k =1,...,n, or mi, i =1,..., 3n
m1 = m2 = m3 m = m = m 4 5 6 etc. Mass-weighted coordinates (for displacements):
qi = √midi (29) Kinetic energy Classical expression:
1 2 1 1 T = m d˙ = q˙2 = p2 (30) 2 i i 2 i 2 i i i i X X X 8 Quantum expression:
1 1 ∂2 T=ˆ ˆp2 = (31) 2 i −2 ∂q2 i i i X X Potential energy:
∂V 1 ∂2V V = V + q + q q + . . . (32) 0 ∂q i 2 ∂q ∂q i j i i 0 i j i j 0 X X X 1 = V + f q + f q q + . . . (33) 0 i i 2 ij i j i i j X X X Forces in equilibrium position are equal to zero:
∂V f = =0 forall i =1,..., 3n (34) i ∂q i 0
V0 is a constant (we will omit it). We use the harmonic approximation: 1 V = f q q (35) 2 ij i j i,j X Matrix - vector notation:
q1 p1 . . f ... f . . 11 1,3n . .. . q = qi p = pi f = . . . (36) . . . . f3n,1 ... f3n,3n q p 3n 3n We use column-vectors, f is the force constant matrix. Now the Hamiltonian may be written: 1 1 H = pT p + qT f q (37) 2 2 We transform the coordinates as follows:
Q = L−1q q = L Q (38)
For the momenta we get:
−1 P = LT p p = LT P (39) 9 Proof (comment by PG): we know that qk = LkmQm. For coordinate Qj the conjugate momentum is (we use the Einstein’s summation convention and the chain rule) 1 ∂ 1 ∂q ∂ 1 ∂ ∂ Pˆ = = k = (L Q ) = j i ∂Q i ∂Q ∂q i ∂Q km m ∂q j j k j k (40) 1 ∂ 1 ∂ = L δ = L = L ˆp = LT ˆp km jm kj kj k jk k i ∂qk i ∂qk Substitution of the new coordinates and momenta into H yields: 1 1 H = P T L−1 L−1 T P + QT LT f L Q (41) 2 2 Let us choose L such that it is real and:
L−1 L−1 T = E (42)
Where E is the identity matrix. Taking the inverse, it follows that:
LT L = E (43)
And for non-singular L it also follows that:
LT = L−1 (44) which means that L is an orthogonal (and unitary, since it is real) matrix. We may further choose L such that:
LT f L =Λ (45)
λ1 0 .. where Λ = . is a diagonal matrix. Multiplying by L gives 0 λ3n f L = L Λ (46)
th For i column of L, denoted by Li, we have:
f Li = Liλi (47) which shows that Li is an eigenvector of f with the eigenvalue λi. Thus to find L we have to diagonalize f, i.e., to find eigenvectors and eigenvalues of f.
10 Substitution of these results into H yields: 1 1 1 1 1 H = P T P + QT Λ Q = P 2 + λ Q2 = P 2 + λ Q2 (48) 2 2 2 i 2 i i 2 i i i i i i X X X In other words the transformation q Q, p P gave us 3n uncoupled harmonic oscillators. The frequencies are→ →
ωi = λi (49) p The Qi are called “normal coordinates”:
Q = L−1 q = L q (50) i ij j ji j j j X X or: Q = L−1q = LT q (51)
We used mass-weighted coordinates qi = √midi, and we can write:
m1 0 1 m /2 2 q = M d with M = . (52) .. 0 m3n 1 1 /2 /2 M has mi on the diagonal. The normal coordinates are then given in terms of atomic displacements by:
1 Q = LT M /2 d (53)
Example: H2O normal modes.
Figure 5: Normal modes of water molecule.
Q — symmetric stretch vibration · 1 11 Q — asymmetric stretch vibration · 2 Q — bending vibration · 3 + 3 overall translations (frequencies = 0)
+ 3 overall rotations (frequencies = 0)
0.3.4 Harmonic vibrations of molecules in cartesian (displacement) coordinates d — coordinates, F — force constants
∂2V 1 ∂2V 1 1 1 F = = m /2 m /2 = m /2 f m /2 (54) ij ∂d ∂d i ∂q ∂q j i ij j i j 0 i j 0 1 1 1 1 F = M /2 f M /2 f = M − /2 F M − /2 (55) The Hamiltonian of the system in simple cartesian coordinates (d and p , d 1 1 ∂ 1 /2 ∂ pd,i = = mi ) is: i ∂di i ∂qi
1 1 (p )2 1 H = pT M −1p + dT F d = d i + F d d (56) 2 d d 2 2m 2 ij i j i i i,j X X To uncouple H we return to normal coordinates Q:
− Q = L 1d d = L Q (57)
−1 p = L T P (58) d Applying this to H:
1 − − T 1 H = P T L 1M −1 L 1 P + QT L T F L Q (59) 2 2 which becomes: 1 1 H = P T P + QT Λ Q (60) 2 2 if:
L T M L = E (61)
12 and
L T F L =Λ (62)
From the last equation we get:
−1 F L = L T Λ (63)
Substituting the previous one gives:
F L = M L Λ (64) so L must satisfy a generalized eigenvalue problem of F with M as the metric (or “overlap”) matrix. 1 1 From the equation F = M /2 f M /2 follows directly the relation between the eigenvectors L and L :
1 1 L = M − /2 L L = M /2 L (65)
1 (It can be also easily seen from q = L Q, d = L Q and q = M /2 d.) Because L−1 = LT we have:
− L 1 = L T M (66) and the normal coordinates are given in terms of atomic displacements by:
− Q = L 1d = L T M d (67) or: L T L Qi = i M d = jimjdj (68) i X L th L where i is the i column of . Remember:
the index i runs over eigenvalues λ and the associated normal coordi- • i nates Qi the index j = kα, k =1,...,n, α =1, 2, 3 (x, y, z) • Thus
Qi = Lkα,imkdkα (69) α Xk X
13 0.3.5 Invariance conditions on the matrix F
We will now derive invariance conditions which show that six eigenvalues λi of the matrix F are equal to zero. The associated normal coordinates correspond to overall translations and rotations of the whole molecule. ∂V 1 ∂2V V = V + d + d d = 0 ∂d i 2 ∂d ∂d i j i i 0 i j i j 0 X X X n ∂V 1 n (70) = V + d + F d d 0 ∂d kα 2 kα,lβ kα lβ α=1,2,3 kα 0 Xk=1 X k,lX=1 Xα,β =0 The force acting on atom k in| the{z direction} α = x, y, z is: ∂V ∂V ∂V n = = + Fkα,lβdlβ (71) ∂di ∂dkα ∂dkα 0 Xl=1 Xβ =0 Translational invariance: this| force{z should} not change if we move all atoms by the same distance: dlβ = δβ for l =1,...,n, hence n n
0= Fkα,lβδβ = δβ Fkα,lβ (72) Xl=1 Xβ Xβ Xl=1 As this should apply to any δ we must require that
n
Fkα,lβ =0 (73) Xl=1 for all k =1, . . . ,n; α =1, 2, 3; β =1, 2, 3(x, y, z) This is the first set of invariance conditions. Note that summation here is over l, with fixed k, α and β. Different k, α and β give different conditions, so indeed we have here a set of invariance conditions. Equation (64) (F L = M L Λ) written in components is:
FijLjr = miLijΛjr = miLirλr (74) j j X X th The matrix elements Ljr for j =1,..., 3n form the r column of L i.e. the th L r eigenvector r n i =(k, α) F L = m L λ (75) j =(l, β) kα,lβ lβ,r k kα,r r Xl=1 Xβ 14 Now suppose that
1 forall l =1,...,n and for β =1 Llβ,r = (76) (0 for β =2, 3
It follows immediately from the invariance condition, eq. (73) that the left hand side of equation (75) equals 0. Thus also the right hand side must be L equal to 0 and, since not all components of r are equal to zero, λr must be equal to zero. So we have found one eigenvalue λr = 0 and the associated L eigenvector r with components = 1 for β = 1 and = 0 for β = 2, 3. ′ ′′ L Similarly we find two more eigenvalues λr = λr = 0 with eigenvectors r′ L and r′′ that have components in the directions β = 2 and β = 3. Let us call these three eigenvalues:
λ1 = λ2 = λ3 =0 (77)
The associated normal coordinates are:
Qr = Lkα,rmkdkα (78) Xkα L Substituting the conditions for r into this general formula leads to:
n
Q1 = mkdk1 k=1 Xn
Q2 = mkdk2 (79) k=1 Xn
Q3 = mkdk3 Xk=1 This can be written in a more compact way:
Q1 n Q = m d (80) 2 k k Q k=1 3 X These are the normal coordinates for the overall translations of the molecule (λ1 = λ2 = λ3 = 0). In fact, they are the coordinates of the center of mass of the molecule. Rotational invariance
15 We will now see how all the atoms move when we rotate the whole molecule and we will again require that the forces on the atoms do not change during this movement.
We rotate the molecule around the axis eγ, γ =1, 2, 3(= x, y, z) over an angle ∆ϕγ . For very small (infinitesimal) rotations it turns out that a vector r0 changes as follows: r = r + ∆ϕ e r (81) 0 γ γ× 0 (This will be demonstrated in chapter 3.) Atom k has equilibrium position ak. After the rotation of the molecule its position becomes r = a + d = a + ∆ϕ e a (82) k k k k γ γ× k Thus the displacement of the kth atom is: d = ∆ϕ e a (83) k γ γ× k The force acting on atom k in the direction α was zero when the atom was in its equilibrium position, and it must still be zero after we rotated the whole molecule (because the relative positions of the atoms did not change): ∂V 0= = Fkα,lβdlβ (84) ∂dkα Xl Xβ After substituting dlβ:
n 0= F ∆ϕ e a (85) kα,lβ γ γ× l β l=1 β X X As this should apply to any ∆ϕγ for γ =1, 2 and 3 we must require that the following invariance conditions are fulfilled:
n F e a =0 kα,lβ γ× l β β=1,2,3 l=1 (86) X X for k =1,...,n,α =1, 2, 3,γ =1, 2, 3(= x, y, z) L L It follows that the vector r with components kα,r = eγ ak α is an eigen- vector of F with eigenvalue λ = 0. There are three such vectors:× γ =1, 2, 3. r Let us label these eigenvectors with r = 4, 5, 6. Associated normal coordi- nates are: n n Q = m e a d = m e a d (87) γ+3 k γ× k α kα k γ× k · k k=1 α=1,2,3 k=1 X X X 16 The vector product satisfies the relation:
(a b) c = a (b c) (88) × · · × Using this relation we get:
n Q = m e (a d ) (89) γ+3 k γ · k× k Xk=1
The vectors e1, e2 and e3 are unit vectors along the axes of the Cartesian coordinate system:
1 0 0 e = 0 e = 1 e = 0 (90) 1 2 3 0 0 1 As a result we have: Q4 n Q = m a d (91) 5 k k× k Q k=1 6 X These are the normal coordinates of overall rotations of the molecule. The associated eigenvalues are λ4 = λ5 = λ6 = 0.
So far we have found six normal coordinates of the molecule (Q1,...,Q6), that correspond with eigenvalues λ1 = . . . = λ6 = 0. The remaining normal coordinates Qi L L T Qi = mk kα,idkα = i M d (92) α Xk X must be found as the eigenvectors of the matrix F , with “overlap matrix” M. For these eigenvectors we have:
L T M L = E (93) where E is the unit matrix. Thus
L T L i M j = δij (94)
The coordinates Q1, Q2, Q3 describe the translations of (the center of mass of) the molecule, the coordinates Q4, Q5, Q6 describe the overall rotations, the remaining Q are the (3n 6) “real” vibrations. We want to separate i − these “real” vibrations from the rotations and translations. This can be achieved by using a “molecule-fixed” coordinate system that rotates and
17 translates with the molecule. Such a system is defined by requiring that at every moment the following conditions are satisfied:
Q1 n Q = m d =0 2 k k Q3 k=1 X (95) Q4 n Q = m a d =0 5 k k× k Q k=1 6 X These are the so-called Eckhart conditions. (Compare to equations (80) and (91).) Setting Q1, Q2, Q3 equal to zero implies that the origin of the molecule-fixed coordinate system is the center of mass of the molecule. Set- ting Q4, Q5, Q6 equal to zero minimizes the coupling between rotations and vibrations. The latter coupling cannot be completely removed, however, in contrast with the coupling between vibrations and translations. All of this will be explained below.
0.3.6 Vibrations in internal coordinates (3n-6)
1. Bond lengths rkl
2. Bending angles Θkln
3. Out-of-plane wagging Θklmn
4. Torsion angles τklmn
Figure 6: Example: Internal coordinates of water molecule (s1 = ∆rkl, s2 = ∆rkm, s3 = ∆Θklm).
18 The Cartesian coordinates rk are functions of the internal coordinates, and this also applies to the displacements dk. There are 3n Cartesian coordinates dk, for k = 1,...,n, but 3n 6 internal coordinates s. Besides the internal coordinates s, there are also− 3 coordinates that describe the position of the molecule in space (actually the position of the center of mass) and 3 angle coordinates that describe the orientation of the molecule.
The force constants Fij of the molecule in Cartesian coordinates have lit- tle physical meaning. Rather, we know the force constants related to bond lengths, bond angles, etc. — the internal coordinates s — these force con- stants form the matrix F :
∂2V F = (96) ij ∂s ∂s i j The harmonic approximation of the potential in the internal coordinates s is: 1 V = sT F s (97) 2 We need to write also the kinetic energy in terms of the internal coordinates s. To achieve this, we first define the transformation s = B d (98)
d is a column vector of length 3n • s is a column vector of length 3n 6 • − B is a matrix of dimension (3n 6) (3n) • − × If s is a small (infinitesimal) change of the internal coordinates, and d is a small change of the Cartesian displacements, then:
∂si Bij = (99) ∂dj
Example H2O - see Figure 6:
s1 = ∆rkl (100) 1 /2 r = (x x )2 +(y y )2 +(z z )2 (101) kl k − l k − l k − l 1 ∂s1 1 − /2 xk xl = [. . .] 2(xk xl)= − (102) ∂xk 2 · − rkl ∂s x x ∂s 1 = l − k = 1 (103) ∂xl rkl −∂xk
19 These elements, together with derivatives with respect to yk, yl, zk and zl, form the 1st row of the matrix B. The remaining elements of this row are equal to zero.
s3 = ∆Θklm (104) rkl rkm cosΘklm = · (105) rklrkm Differentiation would give us the elements of the third row of the B matrix. In a similar way we compute all the elements of the matrix. For the usual types of coordinates 1 — 4 you can find the derivatives in the books of Wilson, Decius, and Cross and of Califano. All derivatives must be calculated for the equilibrium structure of the molecule. B is not a square matrix, so B−1 is not defined. The remedy is to add to s the 6 coordinates Q1,...,Q6 of translation and rotation of the whole molecule. These also are (already presented) linear combinations of d (see section E). After adding these 6 rows B becomes a square and non-singular matrix, so it is possible to write: d = B−1s (106) For the momenta p , conjugate to s we have: s
T p = B−1 p p = BT p (107) s d d s Now we can write the Hamiltonian of the molecule as follows: 1 1 H = T + V = pT M −1p + dT F d = d d 2 2 (108) 1 1 = pT B M −1BT p + sT F s 2 s s 2 where T F = B−1 F B−1. (109) We define the matrix G : G = B M −1BT . (110) Then we can write 1 1 H = pT G p + sT F s (111) 2 s s 2 We want to uncouple the Hamiltonian: 1 1 1 H = P 2 + λ Q2 = P T P + QT Λ Q (112) 2 i i i 2 2 i X 20 To do this we must return to the normal coordinates. We use the transfor- mation: ′ s = L Q (113) Through an analogy to what we have already seen, we derive conditions:
′ − ′ L T G 1L = E (114) ′ T ′ L F L =Λ (115) (116) We can combine these two equations into one:
− ′ ′ 1 − ′ F L = L T Λ = G 1L Λ (117) This is a generalised eigenvalue problem of the matrix F with “overlap ma- − − trix” G 1. As we can see, G 1 is indeed symmetric:
− G 1 = B−1 T M B−1 (118)
After multiplying equation (117) by G from the left side:
′ ′ G F L = L Λ (119) G F L ′ L ′ i = iλi
This is the eigenvalue problem of the matrix G F . This method is called “Wilson’s GF matrix method”. The dimension of this method is 3n 6, as we can omit the 6 rows asso- −−1 −1 T ciated with Qi in B (we don’t need B in G = B M B ) and we define 2 F as F = ∂ V . The internal coordinates do not change when the ij ∂si∂sj whole molecule is translated or rotated. That is, the Eckart conditions are automatically taken into account.
0.4 Rotations of rigid (non-linear) molecules
See also: Paul Wormer, Courses and teaching material http://www.theochem.ru.nl/ pwormer/teachmat.html The rigid rotor in classical and quantum mechanics
21 0.4.1 Rotation matrices and infinitesimal rotations
Choose a coordinate frame that “rotates with the molecule” — such a frame is called a body-fixed (BF) frame. So the orientation of the molecule is defined by the orientation of the BF frame relative to the laboratory = space fixed (SF) frame. Hence, the orientation of the molecule, i.e., the orientation of the BF axes, with respect to the SF frame is described by a 3 3 matrix of direction cosines: the rotation matrix C. × Any orientation (rotation) can be described by three Euler angles ϕ, ϑ, χ:
rotate around the z-axis over ϕ x′, y′, z′ axes (z′ axis = z axis) → rotate around the y′-axis over ϑ x′′, y′′, z′′ axes (y′′ axis = y′ axis) → rotate around the z′′-axis over χ
The corresponding rotation matrices (containing direction cosines) are: cos ϕ sin ϕ 0 cos ϑ 0 sin ϑ − C (ϕ)= sin ϕ cos ϕ 0 C (ϑ)= 0 1 0 z y 0 0 1 sin ϑ 0 cos ϑ − (120) For rotations around the x-axis we have: 10 0 C (α)= 0 cos α sin α (121) x 0 sin α −cos α The Euler (three-angle) rotation matrix is:
C(ϕ, ϑ, χ)= C ′′ (χ)C ′ (ϑ)C (ϕ), (122) z y z which can also be written as (prove this!): C(ϕ, ϑ, χ)= C (ϕ)C (ϑ)C (χ) (123) z y z We define the so-called Lie derivatives or infinitesimal rotations J , J , x y J : z sin ϕ cos ϕ 0 0 1 0 dC (ϕ) − − − J = z = cos ϕ sin ϕ 0 = 1 0 0 z dϕ − ϕ=0 0 00 0 0 0 ϕ=0 (124) 00 0 0 01 J = 0 0 1 J = 0 00 x − y 01 0 1 0 0 − 22 Any vector can be decomposed as: x r = y = xe + ye + ze (125) 1 2 3 z in term of unit vectors: 1 0 0 e = 0 e = 1 e = 0 (126) 1 2 3 0 0 1 The infinitesimal rotations of the vector r are: 0 J r = z = e r x 1 −y × z J r = 0 = e r (127) y 2 x × − y − J r = x = e r z 3 0 × ωx Any infinitesimal rotation ω = ω can be described as y ωz J = ω J + ω J + ω J = ω x x y y z z 0 ω ω − z y (128) = ω 0 ω = Ω z − x ω ω 0 − y x — this is the most arbitrary antisymmetric 3 3 matrix (any 3 3 antisym- metric matrix can be written in this way). × × It is easy to verify that J r = Ω r = ω r (129) ω × Furthermore, it follows that: sin ϕ cos ϕ 0 dC (ϕ) − − z = cos ϕ sin ϕ 0 = J C (ϕ)= C (ϕ)J (130) dϕ z z z z 0− 00 23 0.4.2 The rigid rotor in classical mechanics
A set of particles (atoms) has the coordinates rk(t) in the laboratory (SF) frame. Their masses are mk. The position vector of the center of mass is:
n −1 R(t)= M mkrk(t) (131) Xk=1 with M denoting the total mass, M = k mk. In a coordinate frame which moves withP R(t), but has axes parallel to the axes of the laboratory (SF) frame, the atoms have coordinates:
ρ (t)= r (t) R(t) (132) k k − It follows that: m ρ (t)=0 at each time t (133) k k Xk (prove this). In other words: the position vector of the center of mass is 0 in the center of mass frame. We assume that the particles (atoms) belong to a moving rigid body (molecule). We define a rotating (BF) coordinate frame so that any time t:
ρ (t)= C(ϕ(t), ϑ(t), χ(t))a (134) k k
The vectors ak are the constant position vectors of the atoms in the BF coordinate frame that rotates with the molecule. The angles ϕ(t), ϑ(t), χ(t) are the Euler angles describing the orientation of the BF frame with respect to the SF frame at time t. Finally, we can write:
rk(t)= R(t)+ C(ϕ(t), ϑ(t), χ(t))ak (135) with
r (t) — coordinates of atom k in the laboratory (SF) frame · k R(t) — coordinates of the center of mass in the laboratory (SF) frame · C(ϕ(t), ϑ(t), χ(t)) — rotation matrix containing direction cosines of the · rotating BF axes in the laboratory (SF) frame
24 a — constant position vector of atom k in BF coordinates · k The velocities of the atoms are: ∂r (t) v (t)= k =r ˙ (t) (136) k ∂t k ˙ ˙ r˙k(t)= R(t)+ C(ϕ(t), ϑ(t), χ(t))ak (137) = R˙ (t)+ C˙ (ϕ, ϑ, χ)C−1(ϕ, ϑ, χ)ρ (t) k Rotation matrices are orthogonal:
C−1 = CT (138)
Theorem: The matrix C˙ C−1 = C˙ CT is an antisymmetric matrix. Proof: C CT = E identity matrix T (139) C˙ CT + C C˙ =0 Hence T T C˙ CT = C C˙ = C˙ CT (140) − − Let us assume that:
0 ωz ωy T − C˙ C = Ω = ω 0 ω (141) z − x ω ω 0 − y x Theorem: The relation between the angular velocity ω and the time derivatives ϕ˙ ϕ ∂ q˙ = ϑ˙ = ϑ of the Euler angles is given by: ∂t χ˙ χ ϕ˙ ω = U ϑ˙ = U q˙ (142) χ˙ with 0 sin ϕ cos ϕ sin ϑ − U = 0 cos ϕ sin ϕ sin ϑ (143) 1 0 cos ϑ 25 Proof: C˙ (ϕ, ϑ, χ)= C˙ (ϕ)C (ϑ)C (χ) z y z (144) + C (ϕ)C˙ (ϑ)C (χ)+ C (ϕ)C (ϑ)C˙ (χ) z y z z y z
∂ ∂C (ϕ) ∂ϕ C˙ (ϕ)= C (ϕ)= z = J C (ϕ)ϕ ˙ (145) z ∂t z ∂ϕ ∂t z z
Ω = C˙ CT = J C (ϕ) C (ϑ) C (χ)CT (χ) CT (ϑ) CT (ϕ) ϕ˙ z z y z z y z
(146) + C (ϕ)J C (ϑ) C (χ)CT (χ) CT (ϑ) CT (ϕ) ϑ˙ z y y z z y z
+ C (ϕ)C (ϑ)J C (χ)CT (χ) CT (ϑ)CT (ϕ)χ ˙ z y z z z y z Each “box” yields the identity matrix E (because rotation matrices are or- thogonal), so we have:
Ω = J ϕ˙ + C (ϕ)J CT (ϕ) ϑ˙ + C (ϕ)C (ϑ)J CT (ϑ)CT (ϕ)χ ˙ (147) z z y z z y z y z
Substitution of eq. (120) for the matrices C and C and eq. (124) for J z y z and J , and using the relation in eq. (128) between Ω and ω proves eqs. (142) y and (143). As a result we can express the velocities of the atoms as:
˙ ˙ T r˙k(t)= R(t)+ C C ρ (t) k (148) = R˙ (t) + Ω ρ (t) k One may remember that multiplication of the vector ρ (t) by the antisym- k metric matrix Ω can be replaced by taking the vector product ω ρ (t): × k
r˙ (t)= R˙ (t)+ ω ρ (t) (149) k × k Because ρ (t) is the velocity of atom k in a coordinate frame that moves with k the center of mass, but does not rotate, it follows that ω is the angular velocity in the laboratory (SF) frame
ωSF = ω (150)
26 We may further write
r˙ (t)= R˙ (t)+ ω ρ (t) k × k = R˙ (t)+ C C−1 ω ρ (t) × k (151) − − = R˙ (t)+ C C 1ω C 1ρ (t) × k h i The last line follows from the properties of the vector product under rotation:
C (a b)=(C a) (C b). × × Here we can substitute a = C−1ρ (t) and we get: k k
r˙ (t)= R˙ (t)+ C ωBF a (152) k × k The angular velocity with respect to the rotating (BF) frame is:
ϕ˙ ϕ˙ − − − ωBF = C 1ω = C 1ωSF = C 1U ϑ˙ = V ϑ˙ (153) χ˙ χ˙ with sin ϑ cos χ sin χ 0 − − V = C 1U = sin ϑ sin χ cos χ 0 (154) cos ϑ 0 1 This can be easily shown, if one remembers eq. (143):
0 sin ϕ cos ϕ sin ϑ − − V = C 1U = CT (χ)CT (ϑ)CT (ϕ) 0 cos ϕ sin ϕ sin ϑ = z y z 1 0 cos ϑ = CT (χ)CT (ϑ)CT (ϕ) e , C (ϕ)e , C (ϕ)C (ϑ)e = (155) z y z 3 z 2 z y 3 sin ϑ cos χ sin χ 0 T T T − = C (χ)C (ϑ)e , C (χ)e , e = sin ϑ sin χ cos χ 0 z y 3 z 2 3 cos ϑ 0 1
We have now expressed the velocities of the atomsr ˙k in terms of the velocity ϕ˙ of the center of mass R˙ and the angular velocity ωSF , or ωBF ,orq ˙ = ϑ˙ . χ˙ 27 We want to derive the expression for the kinetic energy, but first we define:
x (t) 0 z (t) y (t) k − k k ρ (t)= y (t) X (t)= z (t) 0 x (t) (156) k k k k k z (t) → y (t) x (t)− 0 k − k k — an antisymmetric matrix with the cartesian (SF) coordinates of atom k. We define a similar matrix for the BF frame: a 0 a a kx − kz ky a = a A = a 0 a (157) k ky k kz kx a → a a −0 kz − ky kx When we consider the elements of the product matrix XT X : k k
XT X = y2 + z2 =(x2 + y2 + z2) x2 k k k k k k k k 11 − (158) T X X = xkyk k k 12 − we see that the matrix
ISF (t)= m XT (t)X (t) (159) k k k Xk is the inertia tensor of the molecule in the laboratory frame. Similarly, we see that: IBF = m AT A (160) k k k Xk is the inertia tensor in the BF frame. It is obvious that IBF is constant in time. The kinetic energy
T = 1 m v2 = 1 m r˙ r˙ = 1 m r˙T r˙ (161) 2 k k 2 k k · k 2 k k k Xk Xk Xk can be written in different ways. In the laboratory (SF) frame we get:
T T T T = 1 m R˙ R˙ + m R˙ ω ρ + 1 m ω ρ ω ρ 2 k k × k 2 k × k × k k k k X X X (162)
28 The second therm vanishes, because
T T T mkR˙ ω ρ = R˙ ω mkρ = R˙ (ω 0)=0 (163) × k × k! × Xk Xk see eq. (133). We may further write that:
Ω ρ = ω ρ = ρ ω = X ω (164) k × k − k × − k and the result becomes:
1 ˙ T ˙ 1 T T T = 2 MR R + 2 ω mkX X ω (165) k k! Xk 1 ˙ T ˙ 1 T SF T = 2 MR R + 2 ω I ω (166) The first term corresponds to the translation of the center of mass, the second to the rotation. As we see, there is an exact separation between translation and rotation. In the BF frame we obtain (also here the cross term vanishes):
T T T = 1 MR˙ R˙ + 1 m ωBF a CT C ωBF a 2 2 k × k × k k X (167) T T = 1 MR˙ R˙ + 1 ωBF m AT A ωBF 2 2 k k k k ! X 1 ˙ T ˙ 1 BF T BF BF T = 2 MR R + 2 ω I ω (168) ϕ Let us now express the kinetic energy in terms of the Euler angles q = ϑ . χ To this end we must substitute: ωSF = ω = U q˙ (169) ωBF = C−1U q˙ = V q˙
Ater this substitution we get:
T T = 1 MR˙ R˙ + 1 q˙T U T ISF U q˙ 2 2 (170) 1 ˙ T ˙ 1 T T BF T = 2 MR R + 2 q˙ V I V q˙
29 which we can also write as
1 ˙ T ˙ 1 T T = 2 MR R + 2 q˙ g q˙ (171) where g = U T ISF U = V T IBF V (172) is the metric matrix (metric tensor — “overlap matrix”) belonging to the Euler angles q (N.B. we have seen this previously, in Wilson’s GF method, where instead of q we used internal coordinates s. The matrix G , defined there, is the inverse − of the metric tensor, G 1 = g . Of course, g and g are different matrices, s s but they play a similar role.) Finally, we present an expression for the kinetic energy in terms of angular momenta J = J = m r r˙ (173) k k k × k Xk Xk In SF coordinates: rk = R + ρ k (174) r˙ = R˙ + ω ρ k × k J = m R R˙ + m ρ R˙ k × k k × Xk Xk (175) + m R ω ρ + m ρ ω ρ k × × k k k × × k Xk Xk Because m ρ = 0, the second and the third term vanish. The fourth k k k term can be rewritten as follows P m ρ ω ρ = m ρ ρ ω = k k × × k − k k × k × Xk Xk (176) = m X X ω = m XT X ω = ISF ω − k k k k k k Xk Xk so that we get J = MR R˙ + ISF ω (177) × ˙ In this equation J CM = MR R is the angular momentum due to the center of mass motion, and ISF ω is× the rotational angular momentum.
30 In BF coordinates: rk = R + C ak r˙ = R˙ + C ωBF a (178) k × k BF BF J = J CM + C I ω Summarizing:
SF SF SF SF J = J CM + J rot J rot = I ω BF BF BF BF (179) J = J CM + C J rot J rot = I ω
Apart from the term related to the motion of the center of mass, we then get for the kinetic energy: − 1 SF T SF 1 SF T = 2 J I J − (180) 1 BF T BF 1 BF T = 2 J I J 0.4.3 The rigid rotor Hamiltonian
To go from classical mechanics to quantum mechanics, we must first write the classical kinetic energy in terms of the coordinates q and conjugated momenta p.
In classical mechanics, if qi are the (possibly curvilinear) cordinates, then the conjugated momenta are defined as: ∂T pi = (181) ∂q˙i
x In cartesian coordinates, r = y , the expression for the kinetic energy is: z 1 2 1 2 2 2 T = 2 mv = 2 m(x ˙ +y ˙ +z ˙ ) ∂T This is consistent with eq. (181), because p = = mx˙, etc., and p = x ∂x˙ mr˙ = mv. ϕ Let us now apply this to the Euler angles q = ϑ , and first derive the χ conjugated momenta p. We return to eq. (171):
1 T 1 T = 2 q˙ g q˙ = 2 gijq˙iq˙j (182) i,j X 31 with the metric tensor given by g = U T ISF U = V T IBF V , see eq. (172). The momenta are: ∂T p = = g q˙ p = g q˙ q˙ = g−1 p (183) i ∂q ij j i j X and the classical Hamiltonian for the kinetic energy is:
T T 1 T −1 −1 1 T −1 T = 2 p g g g p = 2 p g p (184) 1 T −1 T = 2 p g p (185)
According to the postulates of quantum mechanics, the momentum operators conjugated to a set of (curvilinear) coordinates qi are:
h¯ ∂ pi = (186) i ∂qi
The quantum mechanical momentum operators conjugated to the Euler an- gles are: ∂/∂ϕ h¯ p = ∂/∂ϑ (187) i ∂/∂χ From here on, we will use atomic units withh ¯ = 1. We want to quantize equation (185). But we must be careful, because the matrices U and V are functions of ϕ, ϑ, χ, and so is the matrix g. So, p does not commute with g−1. B. Podolski [Quantum-mechanically correct form of Hamiltonian function for conservative system, Phys. Rev., 32, 812 (1928)]: We cannot simply replace the classical momenta in eq. (185) by operators, but we have to use the general formula for the Laplace operator
∂2 ∂2 ∂2 ∆= 2 = + + (188) ∇ ∂x2 ∂y2 ∂x2 in curvilinear coordinates q. This yields:
1 1 1 − /2 T /2 −1 T = 2 g p g g p (189)
32 where g = det(g). We see that, if g did not depend on q, then g would commute with p and we 1 T −1 would have the simpler expression: T = 2 p g p. The expression for g is known in terms of the Euler angles, g = V T IBF V , (Euler angles appear in V , IBF is constant for a rigid rotor). So, we can derive an explicit quantum mechanical formula for T . We start by calculating
T BF 2 BF 2 g = det g = det V I V = det V det I = sin ϑ I1I2I3 (190) I1, I2, I3 are the so-called principal moments of inertia — the eigenvalues of IBF . We can then write:
2 ∂/∂ϕ h¯ ∂ ∂ ∂ −1 −1 T = , , sin ϑ V −1 IBF V T ∂/∂ϑ (191) −2 sin ϑ ∂ϕ ∂ϑ ∂χ ∂/∂χ Further we should plug in the explicit formula for V , eq. (154). It appears that the result can simply be written in terms of the angular momenta J SF or J BF , but we must first quantize these. In classical mechanics: J SF = ISF ω = ISF U q˙ −1 −1 (192) q˙ = g−1p = U −1 ISF U T p
Combining these two equations gives:
−1 J SF = U T p (193)