Time-resolved study of recoil-induced rotation by X-ray pump – X-ray probe spectroscopy

Modern stationary X-ray spectroscopy is unable to resolve rotational structure. In the present paper, we propose to use time-resolved two color X-ray pump–probe spectroscopy with picosecond resolution for real-time monitoring of the rotational dynamics induced by the recoil effect. The proposed technique consists of two steps. The first short pump X-ray pulse ionizes the valence electron, which transfers angular momentum to the molecule. The second time-delayed short probe X-ray pulse resonantly excites a 1s electron to the created valence hole. Due to the recoil-induced angular momentum the molecule rotates and changes the orientation of transition dipole moment of core-excitation with respect to the transition dipole moment of the valence ionization, which results in a temporal modulation of the probe X-ray absorption as a function of the delay time between the pulses. We developed an accurate theory of the X-ray pump–probe spectroscopy of the recoil-induced rotation and study how the energy of the photoelectron and thermal dephasing affect the structure of the time-dependent X-ray absorption using the CO molecule as a case-study. We also discuss the feasibility of experimental observation of our theoretical findings, opening new perspectives in studies of molecular rotational dynamics.

In Secs. I and II below we compute the transition dipole moments of valence photoionization.

PARTIAL TRANSITION DIPOLE MOMENTS OF PHOTOIONIZATION
According to eq. (5) the HOMO 5σ orbital of CO ψ 5σ (r) = n=O,C ψ n,5σ (r − R n ) (S1) is a coherent superposition of the the wave functions of the oxygen ψ O,5σ ≡ ψ O,5σ (r − R O ) and carbon ψ C,5σ ≡ ψ C,5σ (r − R C ) atoms. Here r and R n are the radius vector of the electron and of the n-th atom. We need to compute the transition dipole of the 5σ → ψ k photoionization Now we are in stage to transform the continuum wave function ψ k (r) to the same origin as the atomic wave function ψ n,5σ (r − R n ). First, let us do this using the plane wave approximation ψ k (r) ≈ 1 (2π) 3/2 e ık·r = 1 (2π) 3/2 e ık·Rn e ık·(r−Rn) , which is quite good approximation for X-ray photoionization of the valence shell. In fact, the plane-wave approximation can be strongly improved [1] by replacing the plane wave (2π) −3/2 exp(ık · (r − R n )) by solution ϕ (n) k (r − R n ) of the Schrödinger equations in the vicinity of the n-th atom We assumed in eq.(S2) that ψ k |ψ n,5σ ≈ 0. This is because ψ k |ψ n,5σ ≈ e −ık·Rn ϕ (n) k (r − R n )|ψ n,5σ (r − R n ) ≈ 0. Substitution of the wave functions (S1) and (S4) in eq. (S2) results in the following expression for the transition dipole moment of the valence ionization is the partial transition dipole moment of the ejection of the photoelectron from the 5σ orbital (ψ n,5σ (r)) in the vicinity of the n-th atom. Let us proceed further and write R n in terms of internuclear radius vector This allows to get The opposite signs in these exponents are very important for the Cohen-Fano interference because (d 10 ∝ e ık·R (see Sec. III).

CALCULATION OF d (n)
In this section we show details of derivation of the eq.(23) for d (n) and clarify the meaning of the coefficients A n , B n and C n . Let us choose the molecular frame with the molecular axis along z-axis R z, θ = ∠(k, R). (S9) Using the expansions of the wave functions ψ n (r) and ϕ in terms of the radial integrals P (n) Ll and the spherical functions Y lm (k). It is convenient to use the expansion ofr over the real spherical functions and the relationship between real (Y 1µ (r), µ = x, y, z) and complex (Y 1m (r), m = 0, ±1) spherical function [2]r = µ=x,y,zr Here e µ is the unit vector along the µ-th axis. Putting together this, the expression for the matrix element [2] andk x = sin θ cos ϕ,k y = sin θ sin ϕ,k z = cos θ,R = e z (S14) we obtain the following expression for the transition dipole moment of our interest These components of d (n) and eq.(S14) allows to write the transition dipole moment µ e µ in the invariant form valid in any frame The obtained equation explains the expression (23) for d (n) and clarifies the meaning of the coefficients A n , B n and C n in eq.(23) of the main text (S17)

COHEN-FANO INTERFERENCE
In the present section we explain the partition of the ionization cross-section (4) in three contributions and explain why the interference term σ int (5) is negligibly small in X-ray ionization of valence electrons. Due to the coherence of the oxygen and carbon contributions in the 5σ molecular orbital, one can expect two-center interference of the ψ O → ψ k and ψ C → ψ k ionization channels. Let as compute the 5σ ionization cross section using eq. (S5) Here (but not in the main text) we neglected rather weak dependence onR of the direct terms d (n) 2 in comparison with the strongR dependence of the interference factor exp(ık · R). Thus we get eqs. (4) and (5), where (S19) One can see that the Cohen-Fano (CF) interference term σ int [3] is comparable with the direct terms σ n when the photon frequency is close to the ionization threshold, where sin(kR)/kR ≈ 1 because here the momentum k is small. However, σ int ∝ (kR) −1 is strongly suppressed in the valence X-ray ionization due to the large momentum of the photoelectron and because of random orientation of free molecules.

POLARIZATION TENSOR
To give more insight in the polarization dependence of the probe X-ray absorption (see eqs. (29) and (30) of the main text) here we provide in-deep physical analysis, starting from eq.(24) of the main text: Apparently, J0 (τ ) depends only on the angle θ between the polarization vectors e 1 and e 2 . The reason for this is the random molecular orientation in rotational states J 0 (the formal reason for this is integration overR in matrix elements and summations over all projections of angular momentum M ) and integration over all directions of ejection of the photoelectron. This means that J0 (τ ) exactly coincides with J0 (τ ) averaged over all orientations of the pair (e 1 , e 2 ) with fixed angle θ between them. Following Ref. [4] let us perform this orientational averaging of the product of the cartesian coordinates of the polarization vectors e 1 and e 2 (see Sec. IV A) e 1i e 1j e 2k e 2l = 1 9 δ ij δ kl + (3 cos 2 θ − 1) 5 Here overline denotes the averaging over orientations of the pair (e 1 , e 2 ) with fixed angle θ = ∠(e 1 , e 2 ). As we will see below, the anisotropic term (∝ (3 cos 2 θ − 1) in the polarization tensor (S21) is the reason for the polarization dependence of the probe X-ray absorption (see eqs. (29) and (30) of the main text). Eq. (S21) results in the following expression [4,5] (e 1 ·k) 2 (e 2 ·R) 2 = 1 9 1 + 1 5 The replacement (e 1 ·k) 2 (e 2 ·R) 2 in eq. (S20) by (e 1 ·k) 2 (e 2 ·R) 2 gives the following expression for is nothing special. The same polarization tensor (eqs. (S22) and (S21)) describes the polarization properties of other resonant two-photon processes, for example resonant inelastic X-ray scattering by free molecules [4,5].

Derivation of eq.(S21)
In general case, the polarization tensor e 1i e 1j e 2k e 2l of rang 4 can be constructed as linear combination of three possible combinations of the products of two Kronecker deltas δ ij δ kl e 1i e 1j e 2k e 2l = Aδ ij δ kl + Bδ ik δ jl + Cδ il δ jk . (S24) To find the unkown coefficients A, B, and C we should use the following special sums

Solution of these equations
results in eq.(S21). It is interesting to apply obtained result to the special case of the same polarization vectors e 1 = e 2 = e. Since now cos θ = 1 and A = B = C = 1/15 we get well known result [6] e i e j e k e l = 1 15 (δ ij δ kl + δ ik δ jl + δ il δ jk ). (S27) 5

DETAILS OF THE PROBE SIGNAL CALCULATIONS
In this section we derive eqs. (10) and (18) of the main text. Substitution of the solution (9) in expression (8) for the probability of absorption of the second pulse (see main text) results in the following expression This equation is too cumbersome. We wish to rewrite it in terms of the evolution operators e −ıH1τ and e −ıH2τ which makes expression for σ k (τ, t) not only significantly simpler but also puts forward the dynamics of the nuclear wave packet between the pulses. Using condition of completeness we eliminate the sum over the quantum states λ 1 , λ 2 , λ 1 in eq. (S28) and obtain the expression for σ k (τ, t) in compact operator form (see eq. (10) in the main text).

6
Here Φ(Ω 2 , Γ) = Re Ψ(Ω 2 , Γ) (see eq. (19) of the main text). By integrating σ k (τ ) over the photoelectron momentum k one obtains the absorption cross section of the probe X-ray pulse (eq. (18) of the main text). In agreement with the intuition the dynamics of the nuclear wave packet is fully defined by the evolution e −ıH1τ in the pumped state in the case of short probe pulse. The formal reason for this is that in eq. (S30) e −ıH2τ e ıH2τ = 1, while the physical explanation is that the evolution in the final state does not affect the studied process due to short X-ray pulses (see eq. (15) and related discussion in the main text).

SPHERICAL FUNCTIONS AND CLEBSCH-GORDAN COEFFICIENTS
Here we collect some important equations of the quantum theory of angular momentum [2] used in the main text. To get the final expression (28) for rec J0 (τ ) we use the sum rule for the product of three Clebsch-Gordan coefficients [2] M0M1 C J1M1 J0M0jm C Here we use the conventional notations for Clebsch-Gordan coefficients and 6j-symbols [2]. Let us write down few equations [2] which are needed for the derivation performed in Sec. 2.3.2 of the main text: (e 2 ·R)(e 2 ·R) = 1 3 Using the Rayleigh expansion of a plane wave (eq. (25)) we get (2J 0 + 1)(2j + 1) 4π(2J 1 + 1)