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Response Theory for Linear and Nonlinear X-ray Spectroscopies Exact State Response Theory Patrick Norman Division of Theoretical Chemistry and Biology KTH Royal Institute of Technology, Stockholm A perspective on nonresonant and resonant electronic response theory for time-dependent molecular properties P. Norman, Phys. Chem. Chem. Phys. 13, 20519 (2011) Dalton2015 X-ray absorption spectrum Cu-phthalocyanine Cu B 6 X-ray absorption **DALTON INPUT .RUN RESPONSE **WAVE FUNCTION .DFT CAMB3LYP alpha=0.190 beta=0.810 mu=0.330 **RESPONSE *ABSLRS .ALPHA .DAMPING 0.0045566 # Carbon K-edge .FREQ INTERVAL 10.01 10.04 0.0025 *END OF 4 expt D A 2 0 284 C 285 286 E F 286 287 288 Photon energy [eV] 288 290 289 292 General Frame of Reference I In a simplistic and generalized view of spectroscopy, the observable is the number of detected particles (e.g. photons or electrons) per unit time in a narrow energy interval and into a small solid angle. I The observable is recorded under certain conditions regarding parameters such as temperature, pressure, concentration, and there exist a model (or connection) from which one can deduce molecular properties from the set of measured data. I Under typical circumstances in molecular spectroscopies, these connections can be viewed upon as measures of changes in an observable due to the presence of electromagnetic fields with origins attributed to external sources. I Compared to atomic fields, the electric fields of conventional lasers are relatively weak. A laser delivering pulses of 10 ns duration and 1 mJ in energy and with a spot size of 100 µm produces an intensity of about 0.3 GW/cm2 . This intensity corresponds to an electric field amplitude of some F ! = 5 ⇥ 10 5 a.u., which is several orders of magnitude smaller than the internal electric fields that bind electrons in atomic and molecular systems. Reasons to employ Perturbation Theory Why do we not simply determine time propagate the Schrödinger equation in a direct and nonperturbational manner? I The response functions defined in perturbation theory provide the natural meeting point between experiment and theory with a distinct separation of one-, two-, three-photon, etc., optical processes. I It is a numerically elaborate process to separate out nonlinearities from dominant lower-order components in the polarization. I Only a small number of states (n ⇡ 101 102 ) can be included in nonperturbational approaches. I Calculation of vibrational contributions can hardly be made practical in a nonperturbational approach. Response Theory and Perturbation Theory Response theory may be thought of as a reformulation of standard time-dependent perturbation theory into a form suitable for approximate state theory. Virtually all spectroscopic properties are encompassed by the theory. Possible perturbations are: I time-independent or time-dependent I electric or magnetic I internal or external I geometric distorsions Selection of molecular properties Electric I Polarizability I Hyperpolarizabilities Magnetic I Magnetizability Electric–Magnetic I Optical rotation I Circular dichroism I Faraday rotation I Magnetic circular dichroism Electric–Geometric I IR intensities I Raman intensities I Vibrational polarizabilities I ZPVA polarizabilites Internal fields I g -tensor I Fine-structure I Spin-spin coupling I Shieldings Historical Background Dalgaard & Monkhorst Coupled Cluster Hättig et al. Coriani et al. Sekino & Bartlett Christiansen et al. Koch & Jørgensen ADC Hättig Schirmer Starcke et al. Trofimov & Schirmer Hättig & Hess MP2 Aiga et al. Rice & Handy Runge & Gross Kohn-Sham Deb & Ghosh Zangwill & Soven Multi-Configuration Olsen & Jørgensen Yeager & Jørgensen SOPPA Hartree-Fock Jonsson et al. Hettema et al. Vahtras et al. Oddershede et al. Dunning & McKoy Gao et al. Salek et al. Maitra & Burke van Gisbergen et al. Bast et al. van Leeuwen Norman et al. Packer et al. Norman & Jensen Dalgaard Sekino & Bartlett Saue & Jensen Linderberg & Öhrn General Theory Langhoff et al. 1970 1980 1990 2000 2010 Motivation for derivation of exact state response functions I I I I I I First and foremost, it reveals the dependence of molecular response properties on intrinsic properties of the system, such as excitation energies and transition moments, and it thereby connects di↵erent spectroscopies. It o↵ers the possibility of constructing few-states-models upon which one can base rational molecular design. It provides explicit formulas for response functions in the configuration interaction (CI) approximation. It reveals general properties and symmetries of response functions that are also valid in approximate state theories. It suggests the identification of excited state properties from a study of poles and residues of ground state response functions, e.g., excitation energies are identified as poles of linear response functions. The formulations of quantum mechanics and perturbation theory that are suitable for approximate state theories are best illustrated in exact state theory. A few words about the Schrödinger equation The Schrödinger equation reads as: i~ @ (t) = Ĥ(t) (t). @t The time-independent Schrödinger equation reads as: Ĥ0 n (x) = En (t) = n (x), iEn t/~ n (x)e . The wave function (t) is always time-dependent, but for a stationary state hp̂i = 0 and the density ⇢(t) is time-indepedent. (t0 ) - i~ ˙ = Ĥ (t) - t > t0 Time-evolution for conservative systems; Ĥ(t) = Ĥ0 The evolution of the wave function from some initial time t0 to time t is given by (t) = Û(t, t0 ) (t0 ), where Û(t, t0 ) = e i Ĥ0 (t t0 )/~ . The operator Û(t, t0 ) is referred to as the time evolution operator, or time propagator. Alternatively, the time-dependent wave function can be written in terms of the eigenfunctions of the Hamiltonian: X (t) = cn (t0 ) n (x)e iEn (t t0 )/~ n Note: For cases when the Hamiltonian is time-dependent, a time propagator cannot be constructed in such a simple manner. Molecular properties from the “exact” wave function We will now study a simple system for which we, for an arbitrary external electric field turned on at t = 0, will have access to the exact time-dependent wave function that solves the Schrödinger equation. By exact, we mean ”for all practical purposes exact” and limited only by the numerical representation on the computer, but this aspect is a side-issue to the present discussion. (t0 ) - i~ ˙ = Ĥ (t) - t > t0 This toolbox will let us explore the di↵erent formulations of the theory of molecular properties without relying on perturbation theory. Two-level atom in static electric field (conservative system) In the electric dipole approximation, the Hamiltonian of the system will equal µ̂F 0 Ĥ = Ĥ0 where Ĥ0 is the Hamiltonian of the isolated atom, µ̂ is the electric dipole operator, and F 0 is the amplitude of the external static electric field. E 6 E = Eb Ea Eb b Ea a Ĥ0 = Ea 0 0 Eb µ̂ = 0 µab µba 0 The energies of the ground and excited states are given as the eigenvalues of the Hamiltonian: ⇣ ⌘ det Ĥ ~! Iˆ = 0 The two eigenvalues become Ea + Eb ~! = ± 2 r Ea ) 2 (Eb 4 + (µab F 0 )2 from which, for small fields, the electric-field dependent energies are found to be (µab F 0 )2 (µab F 0 )4 + + ··· E ( E )3 (µab F 0 )2 (µab F 0 )4 Eb0 (F 0 ) = Eb + + ··· E ( E )3 Ea0 (F 0 ) = Ea From these energy expansions we readily determine the electric polarization properties by taking the field derivatives of the energy in the limit of zero field strength. For the ground state, we get µa = ↵a = a = a = @Ea0 @F 0 =0 F 0 =0 @ 2 Ea0 @(F 0 )2 F 0 =0 @ 3 Ea0 @(F 0 )3 F 0 =0 @ 4 Ea0 @(F 0 )4 F 0 =0 =2 (µab )2 E =0 = 24 (µab )4 ( E )3 where we have introduced the electric dipole moment (µ), polarizability (↵), first-order hyperpolarizability ( ), and second-order hyperpolarizability ( ). Induced polarization The permanent dipole moment of the atom is zero, but, in the presence of the external field, there will appear an induced polarization. Polarizability With E = 0.5 Eh and µab = 1.0 a.u., the static ground state polarizability ↵a is equal to 4.0 a.u., and, with F 0 = 5 ⇥ 10 5 a.u., the induced dipole moment amounts to µ0a = 2.0 ⇥ 10 4 a.u. (nonlinear terms are ignored). Charge dislocation The static electric field (with the strength of a conventional laser field) is e↵ectively capable of causing a displacement of an elementary charge in this system by a distance of 2.0 ⇥ 10 4 a.u. Two-level atom in optical field (nonconservative system) In the electric dipole approximation, the Hamiltonian of the system will equal Ĥ(t) = Ĥ0 µ̂F (t) where Ĥ0 is the Hamiltonian of the isolated atom, µ̂ is the electric dipole operator, and F (t) is the external electric field. E 6 E = Eb Ea Eb b Ea a Ĥ0 = Ea 0 0 Eb µ̂ = 0 µab µba 0 Non-stationary states in laser fields Pulse energy Pulse duration Spot size Intensity Electric field b 1 10 100 0.3 4.6 ⇥ 10 5 ω ⇤E = 0.5 a.u. F(t) = F sin(ωt) µ ab= 1.0 a.u. a mJ ns µm GW/cm2 a.u. ψ(0) = ψa i ψ(t) = H ψ(t) H = H0 - µ F(t) Nonconservative system: Study of Induced Dipole Moment By (t) we will denote the time-dependent wave function that is a solution to the Schrödinger equation i~ @ (t) = Ĥ (t). @t We will assume that a nonresonant (! = 0.1 a.u.) external perturbation is slowly swithed on (a = 1/100 a.u.) in accordance with F (t) = F ! sin !t ⇥ erf(at). 10 fs F(t) 5 ×10 −5 0 -5 0 100 200 tim e 300 400 500 Initial Condition The initial condition for our system is that it resides in the ground state prior to exposure of the perturbation and with a phase that is zero at t = 0, i.e. (t) = ae iEa t/~ for t 0 Infinitesimal Time Propagation With a time step t that is small, we can consider the external field to be constant between t0 and t0 + t and thereby get (t0 + t) = Û(t0 , t0 + t) (t0 ) where the time-evolution propagator equals Û(t0 , t0 + t) = e i Ĥ(t0 ) t/~ Time Propagation Repeated application of Û enables us to determine the wave function in the region t > 0, and, given the time-dependent wave function, the dipole moment is obtained as the expectation value of the electric dipole operator according to µ(t) = h (t)|µ̂| (t)i Initialization Time propagation psi[:,0]=[1, 0] for k in range(1,n): psi[:,k]=dot(expm(-1j*(H-mu*F[k-1])*delta),psi[:,k-1]) P[k]=dot(conj(psi[:,k]),dot(mu,psi[:,k])) H=array([[Ea, 0], [0, Eb]]) mu=array([[0, muab], [muba, 0]]) F=Fw*sin(w*t)*erf Populations of States As a measure of the e↵ect of the perturbation on the system, we may study the population in the ground and excited states which we denote by ⇢a (t) and ⇢b (t), respectively. The populations are given by the projections of the wave function on the eigenfunctions of Ĥ0 according to ⇢a (t) = |h a| (t)i|2 Program code popa=conj(psi[0,:])*psi[0,:] popb=conj(psi[1,:])*psi[1,:] ⇢b (t) = |h b| (t)i|2 Induced Dipole Moment Molecular properties are defined by an expansion of the induced dipole moment in orders the time-dependent electric field µ(t) = ↵F (t) + 1 3 F (t) + · · · 6 Polarizability The linear polarizability can be read o↵ directly as the ratio of the amplitudes of the polarization and the electric field ↵= max[µ(t)] ⇡ 4.17 a.u. F! Dispersion The dependency of the molecular property (in this case the polarizability) to the frequency of the perturbation is referred to as the dispersion of the property. Where are the nonlinearities? µ(t) = ↵F (t) + 1 3 F (t) + · · · 6 There are three ways to enhance nonlinear responses: I Design system with large -value I Increase the electric field strength I Tune the frequency to multi-photon resonances Nonlinear responses µ(t) = ↵( !; !)F ! sin(!t)+ 1 ( 3!; !, !, !)F ! F ! F ! sin(3!t)+. . . 6 Fourier transform Let us choose: F ! = 0.02 a.u. and ! = 0.166 a.u.: Rayleigh scattering Third-Harmonic Generation (THG) Dipole moment (a.u.) 0 0.1 0.2 0.3 0.4 Frequency (a.u.) 0.5 0.6 0.1 0 -0.1 0 200 400 600 Time (a.u.) 800 1000 Nonconservative system: expectation value summary I The energy of the system is not well defined as energy transfer occurs between field and system I The dipole moment and other observables (expectation values) are well defined and time-dependent I Molecular properties (response functions) are defined as expansion coefficients of the observable in terms of field strengths I The overall phase of the wave function is of no concern I Linear response functions are directly identified as ratios of induced dipole moments and field strengths (in the small field limit) I Nonlinear response functions can be obtained in the same manner but the procedure is hampered by numerical issues I This time-resolved procedure is applicable to real systems but limited by the need to maintain low matrix ranks Molecular properties from perturbation theory expansions of the wave function Expansions are made in the basis of the exact eigenstates of the zeroth-order Hamiltonian The response theory cookbook recipe 1. Find an efficient parameterization of wave function I I I redundancy state vector normalization phase isolation and secular divergencies 2. Choose an appropriate equations-of-motion I I I based on Schrödinger equation equivalent in exact state theory important in approximate state theory 3. Apply perturbation theory I not much to say, mostly boring work... 4. Form a well-defined quantity of interest, e.g. µ(t), and identify response functions Parametrization by projections (t) = X cn (t) n requires that X n n; cn (t) = hn| (t)i |cn (t)|2 = 1. Parametrization by rotations | ¯(t)i = e e i P̂(t) i P̂(t) |0i; |0i = |0i cos ↵ P̂(t) = X Pn (t)|nih0| + Pn⇤ (t)|0ihn| n>0 i X n>0 sin ↵ Pn |ni ; ↵ ↵= sX n>0 |Pn |2 . Energy 2 1 0 0 0 - 0 - - 0 The Hamiltonian and the Generator-of-Rotations for system equal 0 1 0 E0 0 0 0 P1⇤ Ĥ = @ 0 E1 0 A P̂ = @ P1 0 0 0 E2 P2 0 a three-states 1 P2⇤ 0 A 0 and the energy as a function of transition state amplitudes is E (P1 , P2 ) = h0|e i P̂ Ĥe i P̂ |0i. Electronic Hessian E (P1 , P2 ) = h0|Ĥ|0i + ih0|[P̂, Ĥ]|0i 1 h0|[P̂, [P̂, Ĥ]]|0i + . . . , 2 which gives an expression for the second-order derivatives of the energy (or Hessian) with respect to the transition state amplitides that equals @2E @Pn @Pm = P=0 h ii 1⇣ h h0| |nih0|, |mih0|, Ĥ |0i 2 h h ii ⌘ + h0| |mih0|, |nih0|, Ĥ |0i = 0, @2E ⇤ @Pn @Pm = P=0 h ii 1⇣ h h0| |nih0|, |0ihm|, Ĥ |0i 2 h h ii ⌘ + h0| |0ihm|, |nih0|, Ĥ |0i = (En E0 ) nm . Choice of Equation of Motion I I Rayleigh–Schrödinger @ i~ @t (t) = Ĥ (t); Ehrenfest theorem @ ¯ ˆ ¯(t)i = h (t)|⌦| @t I I Quasi-energy ⇣ Q(t) = h ¯| Ĥ (t) = P n cn (t) n 1 ¯ ˆ ¯ i~ h (t)|[⌦, Ĥ]| (t)i ⌘ @ i~ @t | ¯i Liouville equation @ 1 ˆ]mn @t ⇢mn = i~ [Ĥ, ⇢ mn (⇢mn ⇢eq mn ) Perturbation Theory The not quite so fun part... Hamiltonian and initial conditions The Hamiltonian can be divided: Ĥ = Ĥ0 + V̂ (t); V̂ (t) = X V̂ ! F ! e i!t ✏t e . ! The solutions to the eigenvalue problem of Ĥ0 are known: Ĥ0 |ni = En |ni, where |ni are the exact eigenstates and En the respective energies. Before being exposed to the perturbation, we assume the molecule to be in a reference state |0i—in most cases the molecular ground state—which corresponds to Pn ( 1) = 0 for all n > 0. Ehrenfest theorem The time-evolution of an observable (expectation value) is determined from the Ehrenfest theorem. For a general ˆ we have time-independent operator ⌦, @ ˆ (t)i h (t)|⌦| @t ˆ @ (t)i + h @ (t)|⌦| ˆ (t)i = h (t)|⌦| @t @t 1 ˆ Ĥ]| (t)i. = h (t)|[⌦, i~ Example: For a conservative system, the application of the above ˆ = Ĥ0 gives the result: equation for ⌦ @ h (t)|Ĥ0 | (t)i = 0, @t which shows that the energy is conserved. Response Functions from the Polarization The time-evolution of Pn (t) is determined from the Ehrenfest ˆ = |0ihn|: theorem applied to ⌦ @ ¯ ˆ ¯(t)i = 1 h ¯(t)|[⌦, ˆ Ĥ0 + V̂ (t)]| ¯(t)i. h (t)|⌦| @t i~ Expansion is made with ˆ e i P̂(t) ⌦e i P̂(t) ˆ + i[P̂, ⌦] ˆ =⌦ 1 ˆ [P̂, [P̂, ⌦]] 2 i ˆ + ..., [P̂, [P̂, [P̂, ⌦]]] 6 and ˆ h0|[P̂, ⌦]|0i ˆ h0|[P̂, [P̂, ⌦]]|0i ˆ h0|[P̂, [P̂, [P̂, ⌦]]]|0i as well as h = = = Pn , 0, 4Pn i ˆ Ĥ0 = ~!n0 ⌦. ˆ ⌦, X m>0 |Pm |2 . Perturbation expansion With (1) (2) (3) Pn (t) = Pn + Pn + Pn + · · · , the first-order equation reads @ 1 ˆ ˆ Ĥ0 ]]|0i h0|[P̂ (1) , ⌦]|0i = h0|[P̂ (1) , [⌦, @t i~ 1 ˆ V̂ (t)]|0i. h0|[⌦, ~ or, equivalently, @ (1) Pn = @t 1 (1) i!n0 Pn + hn|V̂ (t)|0i, ~ which, by direct time-integration, yields Z t 1 0 (1) i!n0 t Pn = e hn|V̂ (t 0 )|0ie i!n0 t dt 0 ~ 1 X hn|V̂ ! |0iF ! e i!t = . i~ ! !n0 ! Linear response function ˆ are defined by: The response functions of the observable ⌦ X ˆ ¯(t)i = h0|⌦|0i ˆ + ˆ V̂ !1 iiF !1 e i!1 t h ¯(t)|⌦| hh⌦; !1 1 X ˆ !1 !2 + hh⌦; V̂ , V̂ iiF !1 F !2 e 2!! i(!1 +!2 )t 1 2 + ··· We identify: ˆ ih0|[P̂ (1) , ⌦]|0i = X ! ˆ V̂ ! iiF ! e hh⌦; i!t or, equivalently, ˆ V̂ ! ii = hh⌦; " # ˆ ˆ 1 X h0|⌦|nihn| V̂ ! |0i h0|V̂ ! |nihn|⌦|0i + . ~ !n0 ! !n0 + ! n>0 Two-level atom: response function value Observable: Perturbation: E 6 ˆ = µ̂ ⌦ V̂ ! = µ̂ E = Eb Ea E = 0.5 a.u. Eb b Ea a ↵(!) = µab = 1.0 a.u. ~! = 0.1 a.u. |µab |2 + E ~! |µab |2 = 4.16666... E + ~! Secular divergencies The Rayleigh–Schrödinger expression for the linear response function is: " # ! |0i ! |nihn|⌦|0i X h0|⌦|nihn| ˆ ˆ 1 V̂ h0| V̂ ! ˆ V̂ ii = hh⌦; + ~ n !n0 ! !n0 + ! This response function is not convergent as ! ! 0, and the sum-over-states expression in this form cannot be used in the static limit. Response functions free of secular divergencies are obtained with derivations based on phase-isolated wave functions. Overall permutation symmetry The linear response can be compactly written as ˆ V̂ !1 ii = hh⌦; X P ,1 X h0|⌦|nihn| ˆ V̂ !1 |0i n>0 !n0 ! where ! = !1 is the sum of optical frequencies and P permutes pairs of operators and frequencies: ˆ and (!1 ,V̂ !1 ). ( ! ,⌦) ,1 Poles and residues A residue analysis provides a means to obtain excited state properties from the ground state response function. The poles equal excitation energies and the residues are given by lim !1 ! !f 0 (!f 0 ˆ V̂ !1 ii = h0|⌦|f ˆ ihf |V̂ !1 |0i !1 ) hh⌦; Summary: perturbation theory expansions I Response functions identified from perturbation theory expansions equal sum-over-states expressions I An understanding is provided as to how response properties depend on excitation energies and transition moments, which allows for rational material design I With use of phase isolated wave functions, response functions are free of secular divergences I From residue analyses of ground state response functions we are able to identify excited state properties I A typical calculation of an excitation energy is made by finding a pole of the linear response function (polarizability) rather than an eigenvalue of the molecular Hamiltonian I This approach can be somewhat complicated for system and/or spectral regions showing a high density of states Resonant fields b ω F(t) = F sin(ωt) ⇤E = 0.5 a.u. µ ab= 1.0 a.u. a ψ(0) = ψa i ψ(t) = H ψ(t) H = H0 - µ F(t) What happens if ~! = I E? Response functions of the form 1 X h0|µ̂|nihn|µ̂|0i h0|µ̂|nihn|µ̂|0i ↵(!) = + ~ !n0 ! !n0 + ! n>0 I are not well defined and do not relate to the polarization of the system Perturbation theory is only applicable for a short period of time after the field is turned on Absorption Let us ask the Schrödinger equation: Perturbation theory in the short-time region gives the result: 1 ⇢B (t) = 2 |hf |V̂ ! |0iF ! |2 t 2 ~ Rabi Oscillations Relaxation Sn IC ESA S1 ISC ESA OPA TPA IC T1 Fluoresc. Phosphoresc. S0 Tn Density matrix formalism The density operator is defined as X ⇢ˆ = p(s) | s (t)ih s (t)|, s where the sum denoted a classical ensemble average. If all systems in the ensemble are identical then the summation contains a single term with unit probability. With wave function expansions in the form of projections onto the eigenstates X | s (t)i = cns (t)|ni, n the matrix elements of the density operator becomes equal to X s s⇤ ⇢mn = p(s) cm cn . s Coherence An ensemble of systems with equal state populations, but for which the phases of the wave function components varies in an incoherent manner, will have vanishing elements ⇢mn (m 6= n): ✓ ◆ 1 1 0 ⇢= ⇥ . 0 1 2 An ensemble that can be described by a single wave function, on the other hand, is fully coherent and is said to be in pure state: 1 ⇣ | i = p ei a | 2 ai + ei b | bi ⌘ with a density matrix: ✓ 1 1 ⇢ = ⇥ i( b e 2 e i( a) a 1 b) ◆ . Time evolution and the Liouville equation Let us consider a pure state situation. Then @ @ @ ⇢ˆ = | ih | + | ih | @t @t @t 1 = [H, ⇢ˆ], i~ which is known as the Liouville equation. For all intents and purposes, this equation provides an identical description of the QM system as if one chooses to work with the Schrödinger equation and a wave function approach. The di↵erence lies in the possibility to modify the Liouville equation as to treat e↵ects of spontaneous and collision-induced relaxation and heat bath interactions. Liouville equation with relaxation @ 1 ⇢mn = [Ĥ, ⇢ˆ]mn ⇢eq mn (⇢mn mn ) @t i~ The diagonal elements of the damping parameter matrix will govern the spontaneous population decays: ⌧n = 1/ nn ; n = nn , n 6= 0; 0 =0 we can, for a pure state, draw a conclusion that the o↵-diagonal elements of the density matrix will depend on time according to |⇢mn (t)| = |cm (t)cn⇤ (t)| = |cm (0)cn⇤ (0)|e and we must therefore have mn =( m + n )/2. ( m + n )t/2 , Time propagation of density matrix @ 1 ⇢mn = [Ĥ, ⇢ˆ]mn @t i~ mn (⇢mn ⇢eq mn ) Initialization Time Propagation H=array([[Ea, 0], [0, Eb]]) mu=array([[0, muab], [muba, 0]]) F=Fw*sin(w*t)*erf for k in range(1,n): rho[:,:,k] = rho[:,:,k-1] delta*1.0j*( dot(H - mu*F[k-1],rho[:,:,k-1]) dot(rho[:,:,k-1],H - mu*F[k-1]) ) delta*gamma*(rho[:,:,k-1]-rho_eq) rho_eq=array([[1,0], [0,0]]) rho[:,:,0]=[[1,0], [0,0]] P[k] = dot(rho[:,:,k],mu).trace() popa=rho[0,0,:] popb=rho[1,1,:] ⇢(0) = ✓ ◆ 0 0 ; 0 1 ⇢eq = ✓ ◆ 1 0 ; 0 0 = ⇥ ✓ ◆ 1 1/2 1/2 1 Induced polarization in two-level atom (resonant field, relaxation) = 0.025 a.u. ↵=i max[µ(t)] ⇡ 80i a.u. F! Density matrix from perturbation theory (0) (1) (2) ⇢mn (t) = ⇢mn + ⇢mn + ⇢mn + . . . with (0) ⇢mn = m0 n0 leads to (N) ⇢mn From =e (i!mn + mn )t Z t 1 1 [V̂ , ⇢ˆ(N i~ 1) ]mn e (i!mn + mn )t 0 dt 0 . ˆ (1) = Tr(ˆ ˆ h⌦i ⇢(1) ⌦) we identify the linear response function as " # ! |0i ! |nihn|⌦|0i X h0|⌦|nihn| ˆ ˆ 1 V̂ h0| V̂ ˆ V̂ ! ii = hh⌦; + . ~ n !n0 ! i n0 !n0 + ! + i n0 Two-level atom: response function value Observable: Perturbation: E 6 ↵(!) = ˆ = µ̂ ⌦ V̂ ! = µ̂ E = Eb Ea E = 0.5 a.u. Eb b Ea a |µab |2 + E ~! i~ µab = 1.0 a.u. ~! = 0.5 a.u. ~ = 0.0125 a.u. |µab |2 |µab |2 ⇡ = 80i E + ~! + i~ i~ Complex Polarizability Pole structure Im(z) Re(z) ** * * hhÂ; V̂ iiz = * * ** " # 1 X h0|Â|nihn|V̂ |0i h0|V̂ |nihn|Â|0i + ~ !n z i n !n + z + i n n>0 I Linear response function is convergent for all real frequencies (but will be complex). I No poles in the upper half plane as required for causal propagators. Relaxation in wave function theory 0 E0 0 B 0 E1 + i B Ĥ0 = B . .. @ .. . 0 0 ··· /2 ··· 1 .. . 0 0 .. . ··· En + i 1 n /2 C C C A I Requires a resolution of all states I Non-Hermitian Hamiltonian is not norm conserving Ehrenfest theorem @ ˆ (t)i = 1 h (t)|[⌦, ˆ Ĥ]| (t)i h (t)|⌦| @t i~ ˆ = |nihm|: Apply the Ehrenfest theorem to ⌦ @ 1 h hm| (t)ih (t)|ni = hm|Ĥ| (t)ih (t)|ni @t i~ hm| (t)ih (t)|Ĥ|ni The above equation is thus a mere repetition of the Liouville equation for density matrix element ⇢mn , and a suitable equation-of-motion with relaxation in wave function theory is: @ ˆ nm | i = 1 h |[⌦ ˆ nm , Ĥ]| i h |⌦ @t i~ mn h ˆ nm | i h |⌦ h eq ˆ nm | |⌦ eq i i i Linear response functions Molecular spectroscopies are interpreted as responses to electromagnetic fields. The linear responses of a molecular property to a perturbation are given by hhÂ; V̂ ii! = " # 1 X h0|Â|nihn|V̂ |0i h0|V̂ |nihn|Â|0i + ~ !n ! i n0 !n + ! + i n0 n>0 Selection of spectroscopies properties and associated linear response functions Refraction Absorption Magnetizability Optical rotation Circular dichroism Re Im Re Im Re hhµ̂; µ̂ii! hhµ̂; µ̂ii! hhm̂; m̂ii0 hhµ̂; m̂ii! hhµ̂; m̂ii! Work done by EM fields on particles A point charge in an EM field experiences the Lorentz force: F = qE + q (v ⇥ B) The magnetic force is perpendicular to the magnetic field and the velocity (or current j = qv) of the particle. The latter implies that magnetic forces do not work. The work becomes Z rt Z t Z t Z t 0 0 W = F · dr = F · v dt = qE · v dt = E · j dt 0 We get an “instantaneous power”: dW =E·j dt Linear absorption cross section In the electric dipole approximations, the average change of absorbed energy per unit volume becomes ⌧ d absorbed energy ! = hE(t) · j(t)iT = Im [ e (!)] I dt volume c T where the angular brackets indicate the average over one period of oscillation T . We have used @P(t) + e i!t ); j(t) = @t 1 I = "0 cE02 e (!) = N ↵(!); 2 E(t) = 12 E0 ✏(e "0 i!t We identify the linear absorption cross section as (!) = ! Im [↵(!)] "0 c Summary I Molecular properties are defined by expansions of energy, quasi-energy, polarization, magnetization, etc. in orders of field amplitudes I The diagonal elements of the electronic Hessian contains excitation energies I Use of explicitly unitary parameterizations embeds the requirement of conserving the norm of the wave function I Use of phase-isolated wave functions in expansions yields response functions that are free of secular divergences I Poles and residues of response functions are connected to excited state properties I With spontaneous relaxation, excited state populations may remain small under resonance conditions. The corresponding response functions become complex and describe absorptive and dispersive spectroscopies Some numerical examples with complex response functions Restricted open-shell DFT for radicals **DALTON INPUT .RUN RESPONSE **WAVE FUNCTION .DFT CAMB3LYP alpha=0.190 beta=0.810 mu=0.330 **RESPONSE *ABSLRS .ALPHA .DAMPING 0.0045566 # Carbon K-edge .FREQ INTERVAL 10.01 10.04 0.0025 *END OF X-ray absorption spectrum Cu-phthalocyanine Cu B 6 X-ray absorption Dalton2015 4 expt D A 2 0 284 C 285 Linares et al., J. Phys. Chem. B 115, 5096 (2011) 286 E F 286 287 288 Photon energy [eV] 288 290 289 292 Wave function correlation and self-interaction free spectra Accuracy in absolute transition energies CIS STEX Relaxation: 11.14 eV 1.2 eV CCSD(R3) IPCH2 Expt 285 290 IPCF2 295 Photon energy [eV] 300 305 Fransson, Coriani, Christiansen, Norman, J. Chem. Phys. 138, 124311 (2013) Chemical shifts σ(ω) [arb. units] CCSD level of theory triples corrected energies and scalar relativistic e↵ects included additional common shift of 1.2 eV applied Ethene cis-1,2-difluoroethene Vinylfluoride Trifluoroethene 1,1-difluoroethene Tetrafluoroethene 284 286 288 290 292 294 284 286 288 290 292 294 Fransson, Coriani, Norman, J. Chem.Photon Phys.energy 138, 124311 (2013) PhotonChristiansen, energy [eV] [eV] Silicon L-edge of silane derivatives K- and L-edge X-ray absorption spectrum calculations of closed-shell carbon, silicon, germanium, and sulfur compounds using damped four-component density functional response theory spin-orbit coupling p3/2 p1/2 CAMB3LYP(100%) Fransson, Burdakova, Norman, Phys. Chem. Chem. Phys. (2016) X-ray one- and two-photon absorption XTPA 1b 3p 3s 2a 2p 2s Ne 1a 3d Expt 3d CPP/DFT Ne 3s 2b XTPA CAMB3LYP(100%) 865.0 1s XAS 4p Absorption Cross Section 4p 4s 3p Ne K-edge selection rules 866.0 867.0 868.0 869.0 870.0 Transition Energy (eV) Fahleson, Ågren, Norman, J. Phys. Chem. Lett. (2016) X-ray one- and two-photon absorption XAS Absorption Cross Section Probing the s and p character of the final state Expt 2b2 4a1 CPP/DFT XTPA CAMB3LYP(100%) wikipedia.org 533.0 534.0 535.0 536.0 537.0 538.0 539.0 Transition Energy (eV) Fahleson, Ågren, Norman, J. Phys. Chem. Lett. (2016)