Pure and zero-point vibrational corrections to molecular properties

Transcription

1 Pure and zero-point vibrational corrections to molecular properties Kenneth Ruud UiT The Arctic University of Norway I UNIVERSITETET TROMSØ June

2 Outline Why consider vibrational effects? General principles for treating molecular vibrations The Born Oppenheimer approximation Pure and zero-point vibrational corrections A perturbation theory approach Zero-point vibrational contributions Choice of expansion point Temperature effects on the ZPVs Pure vibrational contributions Examples of pure vibrational corrections

3 Why consider vibrational effects Vibrational spectroscopies provide detailed information about molecular structures Complementary information to electronic spectroscopies Also appears as fine structures to high-resolution electronic spectra Vibrational corrections may be sizable, in particular for molecules with low-frequency modes and properties with strong geometry dependence Some effects for isolated molecules, such as isotope and temperature effects, are purely vibrational in origin Pure vibrational effects may, for static nonlinear optical properties, be more important than the electronic contribution

4 General principles for vibrational effect In general we assume the validity of the Born Oppenheimer approximation The response theory formalism remains valid, but in effect we can resort to (truncated) sum-over-states expressions The nuclei move slower than the electrons, different dynamics Nuclear wave functions are bosonic Several excitation channels available in a single molecule (the different normal modes) Simpler zeroth-order vibrational wave functions are available (the harmonic oscillator) In general (variational) perturbation theory works well, and will be the only approach considered here

5 A brief reminder: The Born Oppenheimer approximation Let us write the total Hamiltonian for the molecule (in a mass-centred frame) as Ĥ tot = ˆT N + Ĥe ˆTN = N 2 2m N 2 N The forces electrostatic and of same order for nuclei and electrons, but nuclei significantly heavier We assume we can solve the electronic Schrödinger equation exactly for any nuclear configuration and obtain the complete set of electronic eigenstates Ĥ e ψ K (r, R) = V K (R)ψ K (r, R) The solution to the molecular Schrödinger equation can thus be expanded in the electronic solutions Ψ(r, R) = Φ K (R)ψ K (r, R) K We could in principle also expand in the set of electronic eigenfunctions evaluated at a single nuclear configuration

6 The Born Oppenheimer approximation Let us consider the action of the nuclear kinetic operator on the molecular wave function ˆT N Ψ(r, R) = N Φ K (R)ψ K (r, R) 2m N N K ] ] ˆTN Φ K (R) ψ K (r, R) + Φ K (R) ˆTN ψ K (r, R) = K 2 N 1 m N ] N Φ K (R) N ψ K (r, R)] ].

7 The Born Oppenheimer approximation Let us consider the action of the nuclear kinetic operator on the molecular wave function ˆT N Ψ(r, R) = N Φ K (R)ψ K (r, R) 2m N N K ] ] ˆTN Φ K (R) ψ K (r, R) + Φ K (R) ˆTN ψ K (r, R) = K 2 N 1 m N ] N Φ K (R) N ψ K (r, R)] ]. The Born Oppenheimer approximation: Assume that the electronic states are well separated and that N ψ K (R) 0; ˆTN ψ K (R) 0 The action of the nuclear kinetic energy thus only acts on the nuclear part of the wave function

8 The Born Oppenheimer approximation cont. The Schrödinger equation for the nuclear motion can thus be written ˆTN + V K (R)] Φ K k (R) = EK k ΦK k (R)

9 The Born Oppenheimer approximation cont. The Schrödinger equation for the nuclear motion can thus be written ˆTN + V K (R)] Φ K k (R) = EK k ΦK k (R) Nuclei move in an effective potential defined by the electronic wave function Implies that the electrons can adjust instantaneously the changes in the nuclear wave function Breaks down when two electronic states are close in energy (conical intersection) Picture taken from /media/riken/research/rikenresearch/figures/hi_3970.jpg

10 More recollection: Solving the nuclear wave function problem Let us perform a Tayler expansion of the potentials of the electrons wrt distortion from equilibrium geometry V (R) = V (0) + i V (1) i i i,j V (2) ij i j ijk V (3) ijk i j k +... At the equilibrium geometry, the gradient is zero Ignore cubic and higher-order contributions The harmonic approximation to the vibrational Schrödinger equation then becomes ˆTN + 1 V (2) ij 2 i j Φ k (R) = E k Φ k (R) i,j Introduce a set of mass-weighted coordinates q i = m i i 2 2 i 2 q 2 i + 1 V (2) q i q j Φ k (q) = E k Φ k (q) 2 q i,j i q j

11 More on the nuclear wave function problem In order to make the equation separable, introduce mass-weighted normal coordinates that diagonalize both the kinetic and the potential energy operators Q i = 3N j=1 L ijq j Ĥ (0) = Q a a 2 a V (2) aa Q 2 a For each normal coordinate, the solution matches that of the harmonic oscillator problem E (0) n = ( n + 1 ) ω ω = k/2 2 Ψ (0) n = N nh n (ξ) exp ( ξ 2 /2 ) ξ = mωq The total wave function a product of the vibrational wave function of the different normal modes, and energy the sum of the individual normal mode energies We recall the properties of the Hermite polynomials Differential equation H n 2xH n + 2nHn = 0 Recursion formula H n+1 = 2xH n 2nH n 1 Orthogonality Hn (x) H n (x) exp ( x 2) dx = 0 n n Normalization Hn (x)2 exp ( x 2) dx = π2 n n!

12 Separation of electronic and vibrational contributions Let us consider the molecular polarizability using a vibronic wf α α,β ( ω; ω) = 0, 0 ˆµα K, k K, k ˆµ β 0, 0 ω K,k (Kk)0 ω + 0, 0 ˆµ ] β K, k K, k ˆµ α 0, 0 ω (Kk)0 + ω K electronic excited states, k vibrational excited states Within the Born Oppenheimer approximation, the vibronic wave functions are separable Ψ K,k = Φ K k (R) ψ K (r, R) The expression for the polarizability is still not separable due to the energy denominators

13 Let us make the approximation ω K,k ω K for K > 0. We can now write α ( ω; ω)) = 0, 0 ˆµα K K ˆµ β 0, 0 ω K0 ω K 0 + 0, 0 ˆµα 0, k 0, k ˆµ β 0, 0 ω k0 ω k 0 + 0, 0 ˆµ ] β K K ˆµ α 0, 0 ω K0 + ω + 0, 0 ˆµ β 0, k 0, k ˆµ α 0, 0 ω k0 + ω In the first expression we have used the closure over k The first contribution is called the vibrationally averaged polarizability (involves an electronic contribution and a zero-point vibrational average contribution) The second contribution is called the pure vibrational contribution ]

14 In short, we can write the total polarizability as the sum of three contributions α ( ω; ω) = α el ( ω; ω) + α ZPVA ( ω; ω) + α pv ( ω; ω) For small molecules, both the pure and zero-point vibrational corrections can be calculated using numerical methods For larger molecules, perturbation theory is the most common approach More accurate methods includes vibrational SCF/CI/CC methods Perturbation theory allows the contributions to be calculated as geometrical derivatives of energies and properties

15 Perturbation theory approach Let us illustrate the approach using ZPV as an example Expanding the vibrational wave function in the expression for the ZPV correction to a property P we have P = n=0 P (n) = n=0 k=0 1 + n λ k Ψ (k) P el λ n k Ψ (n k) ] ( ( 1) m λ l Ψ (l) λ l Ψ (l) ) m l,m=1 We use the harmonic oscillator as reference Hamiltonian and wave functions Ψ (0) (Q) = Φ 0 (Q) = K φ 0 K(Q K )

16 The geometrical perturbed Hamiltonians are then H (1) = F K Q K K H (n) 1 = (n + 2)! KLMN KLM F KLM Q K Q L Q M F KLMN Q K Q L Q M Q N, n 2 The first-order perturbed wave function can then be written Ψ (1) (Q) = K + a 1 K Φ 1 K(Q) + a 3 KΦ 3 K(Q) ] + K L M c 111 KLM Φ 111 KLM (Q) K L b 21 KLΦ 21 KL(Q)

17 We have here introduced the expansion coefficients ( ) a 1 K = 1 F 3/2 K + 1 F KLL 2ω 4 ω L K L 3 a 3 K = 36ω F 5/2 KKK K b 21 c 111 F KKL KL = 4ω K ωl (2ω K + ω L ) F KLM KLM = 12 2ω K ω L ω M (ω K + ω L + ω M ) For zero-point vibrational corrections, the terms b 21 KL are most often ignored and c111 KLM

18 Expansion of the property In a similar manner, we may expand the geometry dependence of the property P P (Q) = P 0 +P 1 +P 2 = P ref + K dp dq K Q K K L d 2 P dq K dq L Q K Q L Using this, we can collect contributions of different orders wrt. the vibrational wave function and the property After some algebra, expressions for the ZPV corrections and pure vibrational contributions can be obtained

19 Zero-point vibrational corrections In general only the three leading-order corrections are included P = Ψ (0) P 0 Ψ (0) + Ψ (0) P 2 Ψ (0) + 2 Ψ (0) P 1 Ψ (1) We may express this also as P = P + K dp Q K + 1 dq K 2 K d 2 P dq 2 K Q 2 K The notation Q K ( Q K 2 ) is here used to denote the vibrationally corrected displacement of the nuclei (or its square) from the equilibrium positions We will see that this expression in general accounts for more than 90% of the zero-point vibrational corrections to diatomic molecules

20 ZPVA continued In terms of the perturbation expansion of our vibrational wave function, we can write P (0) 0 P (0) 2 P (1) 1 = P (1) = = 3N P 4mω K=1 K Q 2 (2) K 3N 6 2 a 1 P K (3) mω K=1 K Q K

21 An alternative expansion point The effective geometry minimizes the energy functional Ẽ = Vel 0 + Ψ (0) H 0 Ψ (0) Equivalent to requiring F eff K L F eff KLL ω L a 1 K = 0 The leading anharmonic correction to the ZPV correction vanish, and higher-order terms can be expected to be smaller Expansion point is isotope and temperature dependent Less accurate results for pure vibrational corrections since only the anharmonicity of the ground state included

22 Accuracy of PT for diatomics HF HCl N 2 F 2 σ F σ H σ Cl σ H σ N σ F σ (1) 1 e σ (0) 2 e σ (0) 4 e σ (1) 3 e σ (2) 2 e σ (2) e Sum σ eff σ e σ (0) 2 eff σ (0) 4 eff σ (1) 3 eff σ (2) 2 eff σ (2) eff Sum

23 Temperature effects We can generalize the ZPV expressions to an arbitratry vibrational state Ψ (0) P 2 Ψ (0) = 1 2 K Ψ (0) P 1 Ψ (1) = 2 2 K ( ) v K ω K 1 dp ak dq K 2 d P dq 2 K ( ) v K ωk Doing a Boltzmann averaging we get for a thermal equilibrium Ψ (0) P 2 Ψ (0) = 1 ( ) 2 1 d P ωk 4 ω K K dq 2 coth 2k K B T Ψ (0) P 1 Ψ (1) = 2 K a 1 K = 1 2ω 3/2 K 1 dp ak dq K ωk F K + 1 ( ) F KLL ωl coth 4 ω L L 2k B T

24 Centrifugal distortions The most important effects of temperature on molecular properties are often due to centrifugal distortions The origin of the centrifugal distortion is the elongation of bonds as higher rotational states are populated at higher temperatures In a semiclassical approach, this contribution is given as Ψ (0) P 1 Ψ (1) = k BT centr 2 K dp 1 dq K ωk 2 α a (αα) K I αα

25 Pure vibrational corrections Let us recall the expression for the pure vibrational correction to the polarizability α pv = 0, 0 ˆµα 0, k 0, k ˆµ β 0, 0 ω k0 ω k 0 + 0, 0 ˆµ ] β 0, k 0, k ˆµ α 0, 0 ω k0 + ω In the case of the zero-point vibrational contribution, we were considering expectation values of the property operators In the case of pure vibrational effects we will consider transition moments (or rather their squared quantities) Pure vibrational corrections are the vibrational response to an external perturbation Could in principle be solved following the approach already described for response theory

26 Pure vibrational contributions for exact vibrational states We note that we can write the pure vibrational contributions to the polarizability more compactly as α ( ω; ω) = k 0 µ 0k µ k0 ω k ω + µ 0kµ k0 ω k0 + ω = µ 2] General observation: Large vibrational dipole transition moment will give a large pure vibrational effect Pure vibrational contributions involve lower-order electronic (hyper)polarizabilities Optical frequencies in general much larger than vibrational frequencies implies they will dominate the denominators Pure vibrational corrections in general small for frequency-dependent properties, but may be substantional for static fields (and also mixed static-dynamic)

27 Pure vibrational contribution to β Consider the expression for β β αβγ ( ω σ; ω 1, ω 2 ) = 1 P σ,1,2 2 np 0 ˆµ α n n ˆµ β p p ˆµ γ 0 (ω n0 ω σ)(ω p0 ω 2 ), Intermediate states can now be either electronic or vibrational Introducing an resolution of the identity for the electron states, we can write the mixed contribution as µα] = 1 ( ) P σ;1,2 (µ α) 2 0k αβ,γ k0 (ω k0 ± ω σ) 1 k 0 For the contribution only involving vibrational states we get µ 3 ] = 1 ( ) P ω;1,2 2 (µ α) 0k µβ k 0 l 0 kl (µγ) l0 (ω k0 ω σ) 1 (ω 1 ω 2 ) 1

28 Pure vibrational contributions by perturbation theory One way to proceed with calculating the pure vibrational contributions follow the approach dicussed for zero-point vibrational wave functions Expand the properties as a function geometry Vibrational wave functions described by (perturbed) harmonic oscillators In principle straightforward, but tedious Creates rather extensive formulas Key quantities: Geometrical derivatives of dipole moments and (hyper)polarizabilities, in addition to cubic (and even quartic) force fields

29 Pure vibrational corrections: α The pure vibrational contribution to the polarizability is given as µ 2 ] (0,0) µ 2 ] (2,0) µ 2 ] (1,1) λ ±ij.. xy.. α v = µ 2] (0,0) + µ 2 ] (2,0) + µ 2 ] (1,1) = 1/2P α,β ω,ω = 1/4P α,β K ω,ω K,L = 1/4P α,β ω,ω K,L,M + F KKL 2 µ α Q L Q M µ α µ β λ ±ω K Q K Q K 2 µ α Q K Q L 2F KLM µ β Q K ω 1 2 µ β Q K Q L ω 1 K λ±ω KL 2 µ α Q K Q L ] K ω 2 L λ±ω M µ β ω 1 K Q λ±ω KL λ±ω M M = ( ω x + ω y +.. ) + ( ω i + ω j +.. )] 1 ( ωx + ω y +.. ) ( ω i + ω j +.. )] 1

30 Pure vib. corrections: β and γ For the higher-order NLO properties, we have β v = µα] + µ 3] γ v = α 2] + µβ] + µ 2 α ] + µ 4] µ 3 ] = µ 3] (1,0) + µ 3 ] (0,1) µ 4 ] = µ 4] (2,0) + µ 4 ] (1,1) For detailed expressions: Bishop and Kirtman, J. Chem. Phys (1991), ibid (1992) We note that the pure-vibrational corrections involve geometry derivatives of lower-order electric properties We also note the need for a large number of geometrical derivatives of electric properties

31 Pure vib.corrections Pure vibrational corrections are in general much more important for static properties than dynamic properties Importance of PV corrections depends on the optical process A very efficient approximation is the infinite-frequency approximation in which all optical frequencies go to infinity. For example, we have γ v ( 2ω; ω, ω, 0) = 1/4 µβ] (0,0) γ v ( ω; ω, 0, 0) = 1/3 α 2] ( (0,0) (0,0) + 1/2 µβ] + 1/6 µ 2 ] (1,0] α + µ 2 ] ) (0,1] α In general, pure vibrational corrections are less important for (electro)magnetic properties due to the absence of first-order magnetic properties

32 Importance of pure vibrational contributions for different processes

33 Vibrational effects: a different view Bond distance and dipole moment in HF as a function of electric field strength R e (F) R e (F =0) R R 0 ( A) µ (a.u.) µ(r e (F),F) µ(r e (F =0),F) F (a.u.)

34 Nuclear relaxation and curvature contributions Using the electric-field dependence of the dipole moment calculated using the field-free equilibrium geometry we thus have α e = d2 E e (R 0, F ) df 2 F =0 Considering instead the field-relaxed molecular structures, we can define the nuclear-relaxation contribution to the polarizability α nr = d2 E e (R, F ) E e (R 0, F )] df 2 F =0 Finally, we have to recall that also the zero-point vibrational energy will have a electric field dependence, giving rise to the curvature contribution α curv = d2 E zpv (R, F ) df 2 F =0

35 How do the two approaches compare? To illustrate the differences between the two approaches, let us first consider the frequency-dependent polarizability α ( ω; ω) α nr = µ 2] (0,0) ω = 0 The differences between the approaches become more interesting for higher-order properties β nr = µα] (0,0) + µ 3] (1,0) + µ 3 ] (0,1) ω 1 = ω 2 = 0 The nuclear relaxation contribution thus includes leading-order anharmonic corrections Arises due to the importance of the cubic force field when changing geometry

36 Curvature contributions We can in a similar manner relate the curvature contributions, and in this manner recover higher-order anharmonicities α curv = α ZPV + µ 2] (2,0) + µ 2 ] (1,1) + µ 2 ] (0,2) ω = 0 Through this approach we thus get a very complete account of anharmonic effects β curv = β ZPV + µα] (2,0) + µα] (1,1) + µα] (0,2) + µ 3] (3,0) + µ 2 ] (2,1) + µ 2 ] (1,2) + µ 2 ] (0,3) ω1 = ω 2 = 0 Determining the curvature contributions computationally is non-trivial

37 PV: Computational considerations The perturbation-theory approach allows for in principle analytic calculations of all properties However, the number of contributing terms are very large Computational complexity of the anharmonic corrections are high (cubic and quartic force fields) The nuclear relaxation approach instead only requires the optimization of the molecular structure in the presence of the field Finite-field approaches does, however, potentially introduce numerical stability problems In principle limited to higher-order processes involving only one optical frequency, although extensions to all-optical frequencies have been introduced Reduced complexity also from field-induced coordinates

38 Some examples of pure vibrational corrections Let us illustrate the importance of the optical process using different polymer chains We report the relative ratio γ v /γ e (0, 0, 0, 0) Dispersion on electronic contribution in general small as we are not close to any poles Hartree Fock data obtained using a 6-31G basis sets Using the inifite-frequency limit Data taken from B. Champagne et al., J. Chem. Phys (2000)

39 Polyynes and polyenes Polyyne chains γ (0, 0, 0, 0) γ ( ω, ω, 0, 0) γ ( 2ω, ω, ω, 0) H-(C C) 4 -H H-(C C) 5 -H H-(C C) 6 -H Polyene chains γ (0, 0, 0, 0) γ ( ω, ω, 0, 0) γ ( 2ω, ω, ω, 0) H-(C=C) 4 -H H-(C=C) 5 -H H-(C=C) 6 -H

40 Cumulenes and methionines Cumulene chains γ (0, 0, 0, 0) γ ( ω, ω, 0, 0) γ ( 2ω, ω, ω, 0) H-(C=C=C=C) 2 -H H-(C=C=C=C) 3 -H Polyene chains γ (0, 0, 0, 0) γ ( ω, ω, 0, 0) γ ( 2ω, ω, ω, 0) H-(CH=N) 4 -H H-(CH=N) 5 -H H-(CH=N) 6 -H

41 γ for larger molecules γ vib Molecule γ(electronic) γ(vibrational) γ vib +γel 100% Hexatriene Octatriene Decapentaene BPLA Sadlej basis set used for (hyper)polarizability derivatives, B3LYP/TZP for the force field

42 Summary of vibrational corrections Some experimental observations can only be explained taking molecular vibrations into account Although in general small, ZPV are crucial to give agreement with highly accurate experimental results, and in some cases even for getting qualitatively correct results Pure vibrational contributions to NLO properties may be larger than the electronic contribution, but their importance depend on the optical process The calculation of vibrational contributions requires in general geometrical derivatives of high-order molecular properties, making these properties computationally expensive