Two-dimensional x-ray correlation spectroscopy of remote core states.

Nonlinear all-X-ray signals that involve large core-atom separation compared to the X-ray wavelengths may not be described by the dipole approximation since they contain additional phase factors. Expressions for the rotationally averaged 2D X-ray photon echo signals from randomly oriented systems that take this position-dependent phase into account for arbitrary ratio between the core separation and the resonant wavelength are presented. Application is made to the Se K-edge of a selenium dipeptide system.

INTRODUCTION

Nonlinear spectroscopy has been widely applied to excitonic systems spanning the vibrational and electronic response of chromophores in biological and engineered systems. X-ray free electron laser sources strong enough to enable nonlinear processes promise to extend these studies to core-excitons in molecules. These short, high-frequency pulses can probe the molecule on the time-scale of the attosecond period of the core-excitation and on the length-scale of the spatial extent of the resonant core orbital. Coherent excitations, which extend over long ranges (compared to the resonant wavelength), lead to violations of the dipole approximation. Quadripole core transitions on a single core are small, because the spatial extent of the core orbital is small compared to the exciting wavelength. Non-dipole transitions between cores can be modeled as dipole transitions multiplied by a complex phase factor dependent on the core location. While rare for IR and UV/Vis active chromophores, this kind of violation is expected to be the norm for strongly coupled, hard X-ray atomic core excitations in molecules. Time-resolved two- and three-pulse X-ray Raman experiments avoid this difficulty by having the field interact twice with each chromophore, negating any dependence on the phase which is associated with the projection of the resonant core along the applied pulse wave vector.

The nonlinear response of quantum molecules to a classical external field can be expressed as a combination of multipoint correlation functions of the matter-field coupling term in the molecular Hamiltonian. These multipoint functions in time and space, depend on both the spatial and temporal phase profiles of the applied field, For typical electronic and vibrational molecular transitions, the applied field wavelength is much longer than any exciton coherence length. In the dipole approximation, the signal is written as the sum of contributions from a dilute collection of molecules. The bulk polarization generated has a spatial dependence on the centers of the molecules, but the spatial phase of the field assumed to be nearly constant for transitions within the molecule . In the optical regime, this phase is important only for chiral signals. For this reason, we will call the part of the signal which is included in the dipole approximation “nonchiral,” and the remainder of the signal the “chiral” part is generally more relevant for X-rays. The lowest terms in the correction for this approximation can be included by a Taylor expansion, This expansion should be avoided when system coherences exist on scales comparable to or larger than the exciting wavelength.

It is also necessary to average the response over the random orientations of the molecules relative to the laboratory frame defined by the wave vectors and polarizations of the applied pulses. Tensor techniques are commonly used to analytically solve for the rotationally-averaged signals for when the matter-field coupling has no spatial phase dependence (i.e., ). In the dipole approximation, a third order four-wave mixing signal will be proportional to the isotropic tensor product , where are the Euler angles between the molecular (a,b,c,d) and laboratory frames of Ref. 17, and denotes the ensemble average over all possible rotation matrices . A core excitonic system including spatial phases with stationary holes will have a signal proportional to the phase rotationally averaged tensor: , where is the wave vector difference between pulses i and j, and R is the position vector difference between cores N and M. An analytic form of the averaged tensor was derived in Ref. 19. This Phased Rotational Averaging (PRA) signal was applied to describe two-photon absorption in excitonic systems. Unlike the signal in the dipole approximation, this expression contains an explicit dependence on the wave vectors of the applied pulses, and the vectors connecting the chromophore positions, relative to the transition dipoles on the chromophores.

In this paper, we apply this averaging to a nonlinear hard X-ray technique. We show that different Liouville space pathways of this phase-dependent signal depend on different combinations of the applied pulse wave-vectors. This extra experimental knob allows groups of pathways to be isolated and measured separately. We consider experiments that are analogs of optical spectroscopy in molecular aggregates, with the molecular chromophores replaced by atomic core orbitals resonant with the X-ray frequencies of the applied fields. They contain terms with transitions resonant with separate cores on a molecule, and phases that depend on the distance between these cores. In nonlinear X-ray spectra, the relevant phases are determined by two length scales: the spatial extent of the core orbital and the distances between resonant cores. If the spatial extent of the core-orbital is small compared to the X-ray wavelength, the system obeys the local dipole approximation. If the spatial phase of the electric field is constant over a volume containing the molecule, the system obeys the global dipole approximation. X-ray pulses are short enough to probe valence electronic motions on subfemtosecond timescales with element-specific selectivity. In many interesting systems, e.g., large conjugated polyenes or transition metal compounds, only the local dipole approximation holds. In this study, we retain a simple single-electron Hamiltonian picture to describe the X-ray response for systems which obey the local but not the global dipole approximation. We introduce the dimensionless parameter , where R is the distance between different cores and λ is a characteristic hard X-ray wavelength. In the global dipole approximation , rotational averaging leads to a contraction over tensor components of the response, the applied polarizations, and an isotropic tensor. This has been widely applied to calculate the nonlinear response of systems of vibrational and electronic excitations.

Rotational averaging over a large collection of transition dipoles including a phase factor representing the position of each core is computationally laborious. Andrews et al. first derived expressions for the fourth order isotropic tensor used for calculating the four-wave mixing signal. They subsequently considered a “phased” rotational averaging tensor, including an exponential containing a dot product between vectors in the molecular and laboratory frames. Expressions for the four-wave mixing nonlinear response in the X-ray regime where are derived using this tensor. The photon-echo signal contains pulse wave vector direction-dependent terms specific to each Liouville space path. In the regime, the nonlinear response is sensitive not only to the linear combination k of the incident pulse wavevectors, but to their individual, component directions as well. The dependence of the photon echo signal in photosynthetic exciton systems on the applied field wave vectors and the polarizations of the applied pulses has been investigated by Mercer et al. We adopt the same typical boxcar geometry. We apply these results to a model system of a selenium dipeptide, with R = 5.803 Å and λ = 0.979 Å, making . The signal is shown to contain a factor dependent on both the polarizations and the directions of the wave vectors of the applied pulses. Numerical results are then presented.

THE PHOTON ECHO SIGNAL FOR ARBITRARY κ

First, we discuss the general case by including a spatial phase in the photon echo signal for an aggregate of coupled two-level systems, showing an analytic solution exists for the collinear experiment, in which the applied pulses have the same wave vector. Then, we present the X-ray experiment for noncollinear pulses. Expressions for the X-ray photon echo signal were derived in Ref. 22. We consider an electronic Hamiltonian partitioned between a material part defined in terms of the orbital creation and annihilation c operators, with orbital energies , coupled by the Coulomb interaction The nuclei are considered fixed in this treatment. The explicit time-dependence of the Hamiltonian in Eq. (3) is through an externally applied classical field The nonlinear response formalism casts the signal as the heterodyne-detected overlap between the field emitted by the material polarization generated by the first three pulses overlapped with a fourth pulse. The electric field is written with polarization vectors e, envelopes , arrival times , and central frequencies . In the frequency domain, this expression becomes

The dipole operator coupling the core-orbital to the field also contains a spatial dependence on the location of the chromophore R, which is a two level subunit in the aggregate, or a resonant core in the X-ray experiment. The dipole operator in the orbital basis is defined as where The eigenstates of are the ground and valence excited states , are singly core-excited states with holes on the N or M core, respectively, , and the doubly core excited states (see Fig. 1). We make the local dipole approximation described in the introduction by assuming the matrix elements of the dipole operator are taken relative to the position of the chromophore, with the phase of the jth electric pulse at that chromophore.

FIG. 1.

Level diagram for the valence, single and double core-excited states of the selenium dipeptide. Double-arrows mark dipole coupled states. The wavelength resonant with the Se K-edge transition (12658.00 eV) is 0.979 Å, and the Se atoms are separated by 5.803 Å (5.924 ).

The general expression for the PRA signal, after accounting for a spatial phase, and an arbitrary rotation of the molecular system by an Euler rotation matrix , modifies each dipole, , leading to the following tensor average: In the photon echo experiment, k = –k1 + k2 + k3. Substituting this into Eq. (13), we get The indices refer to tensors defined in the molecular (laboratory) frame of reference. are matrix elements of the Euler rotation matrix, and is the ensemble average over the relative orientations of these two frames. If Eq. (14) can be placed into a form with a single dot product in the exponential, the formalism given in Ref. 19 can be applied. Two different ways are now presented to perform this simplification.

PHOTON ECHOES IN A COLLINEAR PULSE CONFIGURATION

The collinear pulse geometry is the simplest; we now apply the non-dipole averaging to a collinear photon-echo signal on a model aggregate of coupled two-level systems (see Fig. 2). Phase-cycling is a technique used to extract an equivalent to the photon echo by varying the phases between pulses. Originally used in nuclear magnetic resonance (NMR), it has been adapted to multidimensional optical experiments. If , then the dot products in Eq. (14) can be combined.

FIG. 2.

System of coupled two-level systems.

We define the excitonic dipole operator and the single and doubly excited states in terms of local chromophore excitation creation and annihilation (B) operators. The eigenstate dipole operators are expanded in the local state basis using this notation. The response contains spatial phase factors that depend on the chromophore positions (see Fig. 3), and an additional sum over local sites , and . The signal is a sum of three terms The ground state bleaching, and stimulated emission term, contain only singly excited states. The expression for the excited state absorption The tensor averages in these expressions can be evaluated using the results of Appendix B. Only homogeneous broadening was considered in the derivation of Eqs. (19)–(21).

FIG. 3.

Ladder diagrams and phase factors contributing to the response for the coupled two-level systems, corresponding to Eqs. (19)–(21).

The generic excitonic systems described above have delocalized eigenstates, and the four pulses can interact at four points in space (R, , R, in Fig. 3). Core holes, in contrast, are localized and the applied pulses must create and destroy a hole on the same site. The signal thus only involves two cores (R). One of the two selenium atoms in Fig. 1 is closer to the peptide nitrogen than the other, resulting in a slightly different chemical environment. We assume the through-space coupling between the core holes is much smaller than their orbital energy splitting, leading to a nearly stationary core-hole. A thorough treatment of the core-hole hopping dynamics between equivalent cores in identical chemical environments has yet to be performed, and will be the focus of future studies. The core-excitations have the form of excitations from stationary cores to diffuse frontier orbitals, forcing each participating core to absorb and emit with the X-ray field, and making the factors in Eqs. (19)–(21) vanish and the dipole approximation holds when the wavevectors of all four pulses are the same. The form of the X-ray excitations, with stationary core-holes is the reason the four-point diagrams in Fig. 3 simplify to the two-point diagrams given in Fig. 4. For collinear pulses, the photon echo X-ray signal carries no dependence on the intercore separation, only when the applied pulses have different wavevectors will the signal contain a dependence on the intercore distance.

FIG. 4.

Ladder diagrams for the photon echo signal, including the wave vector dependent phase, corresponding to Eqs. (23)–(26).

X-RAY PHOTON ECHO FOR NONCOLLINEAR PULSES

We now describe the X-ray photon echo signal for noncollinear pulses. Ladder diagrams for the X-ray photon echo signal are given in Fig. 4. Unlike the coupled, two-level system case, each excitation is associated with its own core, which we label with subscripts . The total signal is the sum of four contributions the ground state bleaching the excited state emission and excited state absorption with two orderings of core-excitations With heterodyne detection, it is possible to measure the real and imaginary parts of the generated field by interference with the last field. We assume we are measuring the real part of the signal. The lineshape functions the frequency domain form of the electric pulses and the perturbative factor contribute to the factor in each expression. A resonant contributions to the fourth-order signal only include terms where the number of core-excitons created equal the number destroyed. We neglect core-hole migration, so that only two cores contribute to each term. Equations (22)–(26) will now be applied to calculate the signal for a model selenium dipeptide.

APPLICATION TO SELENIUM DIPEPTIDE

Charge and energy transfer in protein systems is a field of growing technical interest as these systems are currently more efficient than artificially designed catalysts and solar cells. Selenium is isoelectronic with oxygen and has been used to tag proteins and oligonucleotides for determining their structure using the multiwavelength anomalous dispersion (MAD) technique. Recently, small oligotides were synthesized with selenium substitute directly for the oxygen on the amide backbone. As this backbone is often implicated in charge transfer in proteins, a selenium dipeptide system (Fig. 1) is an attractive system to investigate these nonlinear X-ray effects. The core-excited states of the selenium dipeptide (see Fig. 1) are treated using the orbital approximation and neglecting core-relaxation. The X-ray absorption near edge structure (XANES) in this approximation is shown in Fig. 5, with the applied pulse power spectrum in blue. An active orbital space of the four virtual and four occupied orbitals are taken and used to approximate the core singly and doubly core excited states. The valence states are also treated using the orbital approximation . All orbitals, energies, and dipole matrix elements were calculated using the Gaussian 03 code, using the B3LYP functional and the aug-cc-pVTZ basis set. Each peak is broadened by a Lorentzian with a 2.20 eV FWHM representing the core-hole lifetime. The level of theory is crude, as demonstrated by the large energy shift between the calculated core-edge in Fig. 5, and the experimental 12.65 keV Se K-edge, but is expected to qualitatively reproduce the XANES spectra.

FIG. 5.

Se K-edge XANES spectra of the selenium dipeptide system. Blue line is the power spectrum of the pulse used for the 2D signals.

We assume a boxcar pulse geometry (see Fig. 6), with an apex angle α of 30∘. This is larger than the small solid angle usually chosen for experimental convenience. The apex angle in the boxcar geometry (see Fig. 7) fixes the size of the wavevector differences that contribute to κ; by increasing the difference between the wave vectors, the non-dipole approximation contribution to the signal will be larger. We define “out” polarizations, which are included in a plane containing the center of the pyramid and the wave vector, and “about” polarizations, which are perpendicular to both the wave vector and the “out” polarization (see Fig. 7). Following Appendix B, the signal is a linear combination of nineteen components, weighted by the spherical Bessel function . We choose the all “out” polarization with a boxcar geometry with an apex angle of 30∘. The result is shown in Fig. 8. We find that the nonchiral part of the signal—the signal calculated using the dipole approximation—is roughly three orders of magnitude smaller than the chiral signal. This could be predicted from the geometry of the selenium atoms, as shown by the vertical line in Fig. 9, the first order contribution to the response is expected to be weak.

FIG. 6.

Boxcar pulse geometry with apex angle α, field wave vectors and polarizations .

FIG. 7.

Polarizations defined in boxcar geometry. shares the plane containing the axis of the pyramid and .

FIG. 8.

(Left) Nonchiral (with 60 contours spanning ±3 × 10−4) and (right) chiral (60 contours spanning spanning ±1) contributions to the real part of the PRA signal. Notice that the chiral signal is much stronger.

FIG. 9.

Spherical Bessel functions , with the inter-atom Selenium distance (in units of Å) given by the vertical line.

The PRA signal can be written as a linear combination of components, which is written conveniently written in block matrix form, with five components given in Appendix B. Each block is weighted by a spherical Bessel function with an argument . When the resonant core wavelength (λ) is much larger than the intercore distance (R), only is nonzero, and only the first block corresponding to the dipole allowed transitions (V0) contributes to the signal. The signals are broken down into their contributions from each matrix element of the V vector in Figs. 10–14. Although these may not be measured separately, they provide a framework for interpreting the overall signal given in Fig. 8. The nonchiral signal in Fig. 8 is the sum of the three signals in Fig. 10, the nonchiral part is the sum of the remaining contributions in Figs. 11–14. The nonchiral signal is dominated by the last signal from Fig. 14, and we include the rest to show there is additional information in these signals. The first block of V in Fig. 10 represents the dipole-approximated signals and are fairly low in amplitude compared to the rest of the signals, as predicted above. The next block in Fig. 11 contains several contributions with a comparable intensity to the nonchiral signal, but with opposite and canceling phases ((iv) and (v); (viii) and (ix)). The unique contributions (vi) and (vii) are much lower in amplitude, and contribute negligibly to the final signal. The third block contains several contributions with strong and noncanceling weights, taken together the signals (x)–(xiii) should contribute at ∼1% of the total signal intensity, which can be compared to the signal very intense (xvi) contribution. The most intense contribution to the signal is in the last matrix element, shown in Fig. 14.

FIG. 10.

Contributions from the chiral parts of the V vector [Eq. (B1)], the first three matrix elements. There are 30 contour lines (i), (ii) from ±7 × 10−5, and (iii) from ±5 × 10−4.

FIG. 14.

Contributions from the last and most intense nonchiral matrix element of the V vector, scaled to ±1.

FIG. 11.

Contributions from the second block, nonchiral parts of the V vector [Eq. (B1)], the contour ranges are ±2 × 10−4, ±2 × 10−4, ±3 × 10−7, ±1 × 10−7, ±1.5 × 10−5, ±1.5 × 10−5, respectively.

We next turn to signals that can be experimentally discriminated, by varying the polarizations of the measurement. There are 24 = 16 choices of polarizations for each choice of apex angle in the boxcar geometry. The different signals for are shown in Fig. 15. The signals form two groups with minor changes in the lineshapes and sign changes. Spectra in which the polarizations are paired (that have two “out” polarizations and two “about” polarizations, as described in Fig. 7) form one group, the other group has one polarization of one type, and the rest of the other. For an apex angle near zero, the chiral XXXY signal would be in the second group, and the nonchiral XXYY, XYXY, and XYYX signals would be in the first.

FIG. 15.

Polarization dependence of signals with the boxcar geometry. Polarizations are defined in Fig. 7.

CONCLUSIONS

The selection rules for the X-ray photon echo signal make the collinear signal independent of intercore distances for both homonuclear and heteronuclear experiments. For homonuclear experiments with variable wave vectors, the spatial phases corresponding to the core positions have to be taken into account. We have simulated the photon echo signal, including a rotationally averaged complex exponential spatial phase corresponding to each core position without recourse to a Taylor expansion in this phase. This formalism was applied to a selenium dipeptide, where the selenium atoms were separated by a convenient distance, in which the nondipole contributions to the signal were expected to be large (see Fig. 9). Similar situations will be encountered frequently in hard X-ray nonlinear spectroscopy of the heavier elements.

With our choice of pulse bandwidth, and the assumption of four active virtual and occupied orbitals, there are 16 states in the and bands, and 8 states in the singly excited band, leading to 4096 combinations contributing to the signal. Individual contributions from core-excitations to different virtual orbitals are largely obscured by broadening due to the short core-hole lifetime at hard x-ray excitation energies. It may be possible to fit the interatomic distance between the cores by simulation, although it would be difficult to imagine a model compound that smoothly interpolates this distance in steps small compared with the X-ray wavelength.

The present results can be extended in various ways. The general excitonic response with collinear pulses should be explored, especially comparing the weak, chiral signal calculated using Taylor expansions to the full analytic treatment described here. Also, for the X-ray experiment, we have noted but not exploited the fact that different diagrams depend on different wave vector differences. By taking the numerical derivative of the signal with respect to the wave vector directions, and holding the other two wave vectors fixed, we could isolate peaks from only one pair of diagrams or the other, as in Ref. 21. This should extend the chiral discrimination of the X-ray signal described here.