One-dimensional cuts through multidimensional potential-energy surfaces by tunable x rays

Sebastian Eckert,1,2,* Vinícius Vaz da Cruz,3,† Faris Gel’mukhanov,3,4 Emelie Ertan,5 Nina Ignatova,3,4 Sergey Polyutov,4 Rafael C. Couto,3 Mattis Fondell,2 Marcus Dantz,6 Brian Kennedy,2 Thorsten Schmitt,6 Annette Pietzsch,2 Michael Odelius,5,‡ and Alexander Föhlisch1,2 1Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Strasse 24-25, 14476 Potsdam, Germany 2Institute for Methods and Instrumentation for Synchrotron Radiation Research FG-ISRR, Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein-Strasse 15, 12489 Berlin, Germany 3Theoretical Chemistry and Biology, Royal Institute of Technology, 10691 Stockholm, Sweden 4Institute of Nanotechnology, Spectroscopy, and Quantum Chemistry, Siberian Federal University, 660041 Krasnoyarsk, Russia 5Department of Physics, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden 6Research Department Synchrotron Radiation and Nanotechnology, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland


I. INTRODUCTION
Multidimensional potential-energy surfaces (PES), dynamic pathways, and reaction coordinates are powerful conceptual tools for molecular science and chemistry.Experimental reconstruction of multidimensional PES-spanning from equilibrium to strong geometric distortion-of coupled degrees of freedom poses a challenge.Ideally, knowledge and interpretation of the full experimental vibrational spectrum are needed to solve this inverse problem [1][2][3][4][5].It is almost impossible to extract such information from experimental spectra of large polyatomic systems.However, knowledge of the full multidimensional PES is redundant in many situations.Indeed, the majority of practical applications requires knowledge of the PES along directional reaction pathways, such as in imaging of folding energy landscapes by single-molecule force spectroscopy [6,7] instead of the full PES landscape along all degrees of freedom of the investigated system.
We present subnatural linewidth resonant inelastic x-ray scattering [8][9][10][11] (RIXS) as an experimental approach that delivers this directional reconstruction method of the groundstate potential-energy surface.The strong site and state sensitivity of RIXS uniquely enables probing the electronic groundstate PES in different directions using resonant x-ray scattering through selected intermediate core-excited electronic states.The role of these intermediate states is crucial for the potential reconstruction, because it defines the selective propagation of the nuclear wave packet in state-and site-specific directions.Indeed, the core-excited-state wave packets propagate along the coordinates defined by the orientation of valleys of the selected core-excited-state PES.The projections of these directional wave packets onto vibrational states of the electronic final state, in the emission step of the RIXS process, give access to the final-state PES along the selected pathway in a wide range of geometry distortions (Fig. 1).
Quite often, vibrational states are clustered into groups of close-lying vibrational levels (typically unresolved in current RIXS spectra) [12][13][14][15].In principle, the fine structure of each group can be resolved using RIXS of super-high resolution.However, we face here a nonintuitive situation where the knowledge of such a fine structure of overlapping vibrational states is redundant and even constitutes an obstacle in our reconstruction technique.Instead, we need only much simpler information, namely, the centers of gravity of each group, which form a pseudospectrum that is unique for each propagation direction of the wave packet on the core-excited PES.We have found that this pseudospectrum allows us to extract one-dimensional cuts of the electronic ground-state potential along the propagation direction of the nuclear wave packet in the selected core-excited state.

II. RESONANT INELASTIC X-RAY SCATTERING OF WATER MOLECULES
To exemplify the PES reconstruction, we study the electronically elastic RIXS process in water molecules, which ends up in the electronic ground state.The O 1s x-ray absorption spectrum of gas-phase water exhibits three wellseparated absorption resonances [11,16] which correspond to excitations of the O 1s electron into the unoccupied molecular levels 4a 1 , 2b 2 , and 2b 1 .The theoretical simulations of the core-excited-state wave-packet propagation in Fig. 1 show that the nuclear wave packet moves along different state-specific reaction coordinates on the different core-excited states within the lifetime of the O 1s core hole.The wave packets, confined within the PES valleys, are localized along the OH bonds of the dissociative |O1s −1 4a 1  1 state, between the OH bonds for the |O1s −1 2b 1 2 potential, and exclusively along the bending coordinate for the |O1s −1 2b 1  1 state [11].Upon decay, these directional wave packets are projected onto the vibrational levels of the ground state, thus enabling reconstruction of different cuts through the PES (Fig. 1).
Here, we consider the RIXS spectra of H 2 O for photon energies tuned in resonance with the |O1s −1 4a 1  1 , |O1s −1 2b 1 2 , and |O1s −1 2b 1 1 transitions.The wave packets presented in Fig. 1 were used in Refs.[11] and [17] to compute the RIXS spectra at each absorption resonance and lead to good agreement with experiment.The exclusive presence of stretching excitations in the |O1s −1 4a 1  1 spectrum, dominance of stretching excitations with presence of a single bending overtone in the |O1s −1 2b 1 2 spectrum, and the exclusive existence of bending excitations in the |O1s −1 2b 1  1 spectrum reflect the coupling of the characteristic confinement of the core-excited-state wave packets to the experimentally detected ground-state vibrational modes [17,18].

III. EXPERIMENT
The RIXS end station of the ADRESS beam line [19] at the Swiss Light Source was utilized to acquire the experimental spectra with a resolution of 75 meV.The H 2 O (g) sample, generated by evacuation from a ∼10 ml liquid sample reservoir at a temperature of 60 • C, was transferred through previously evacuated and heated steel capillaries to the interaction region under permanent evacuation.In this way a constant flow and thus continuous sample replacement was achieved.At the point of interaction with the synchrotron x-ray beam, the sample was separated from the UHV experimental chamber by a 150-nm thin silicon nitride membrane.The surface of the membrane was positioned under an angle of 45 • with respect to the incoming x-ray beam, allowing for a transmission of both the incident as well as the emitted photons in a 90 • scattering geometry.
The energy calibration of the ADRESS beam line was determined according to the absorption maximum at the π * resonance of O 2 gas at 530.519 eV to be in agreement with the data presented by Saitoh et al. [20], who reported 530.521 ± 0.01 eV.Disagreements about this absolute energy exist in literature as pointed out by Prince et al. [21].The nominal energies according to the beamline calibration for the spectra recorded at the different resonances are 533.509eV for |O1s −1 4a 1 1 , 535.289 eV for |O1s −1 2b 1 2 , and 536.699 eV for |O1s −1 2b 1 1 .From the comparison between the incident-energy-dependent experimental spectra and simulated spectra considering the detuning dependent core-excited-state dynamics we estimate the detuning from the different resonances to be 0.05 eV for |O1s −1 4a 1  1 , −0.025 eV for |O1s −1 2b 1 2 , and 0.2 eV for |O1s −1 2b 1  1 [17].

IV. ELECTRONIC STRUCTURE METHODS AND RIXS THEORY
To compute the potential-energy surfaces of the ground and core-excited states we employed the MOLCAS 8.0 package [22] using the scalar relativistic restricted-active-space selfconsistent field (RASSCF) method [23] followed by secondorder perturbation theory (RASPT2) method [24].The atomic orbitals were described by an ANO-RCC [25] basis set in combination with a (2s2p1d) Rydberg basis similarly to Ref. [26].The normal vibrational modes of the ground state were obtained at the CASPT2 (8,9) level.An active space with 10 electrons consisting of 11 orbitals in RAS2 and RAS3 was used in the RASPT2 calculations (in C s symmetry) of the potential-energy surfaces.To reach both ground and coreexcited states, separate RASSCF calculations were performed with double or single occupation of O 1s (which is frozen from the Hartree-Fock calculation) in the RAS3 subspace.RASSCF state-averaging and multistate RASPT2 were carried out for |O1s −1 4a 1 1 and |O1s −1 2b 1 2 .We compute the RIXS cross section assuming that the stretching and bending are uncoupled (so-called 2D+1D model [11,17] using a time-dependent representation as the function of the energy loss ω − ω .Here ω is the frequency of the scattered photon, (0) and m are the total zero-point energy and the bending vibrational energy of the ground state.To find the autocorrelation function, where = ω − ω c0 + (0) is the detuning of the incoming photon frequency from the frequency ω c0 of the vertical transition 0 → c.
are the coordinates of the potential minima of the ground and core-excited three-dimensional (3D) potentials.

V. 1D CUT OF THE PES AND THE PSEUDOSPECTRUM
Let us illustrate, based on the detected RIXS transitions, how the confined core-excited-state wave packets can be used to reconstruct one-dimensional (1D) cuts through the ground-state PES.To give insight into the physics, it is enough to consider only the coupled stretching dynamics.The two-dimensional (2D) Hamiltonian of the coupled stretching motion of the ground electronic state in its general form Here μ i are the reduced masses for motion along the general coordinate Q i .The kinetic energy operator is not diagonal due to the mass polarization term ∝ ζ .It is known that the eigenvalues n 1 n 2 of the ground-state 2D Hamiltonian h 2 form groups [15,17,18] according to the group number Each nth group consist of n + 1 close-lying vibrational sublevels as illustrated in Fig. 2. Such clustering of the vibrational states into groups occurs in many molecules containing H-X bonds (e.g., H 2 X, H 3 X, H 4 X) [11,15].In our measurements the fine structure of each nth group is not resolved and each nth peak is instead characterized by its center of gravity, which is sensitive to the direction of localization of the nuclear wave packet in the core-excited state, as seen in Fig. 2. Indeed, the probability P (n) n 1 n 2 is defined by the square of the scattering amplitude [8,17,18], which is the projection of onto the vibrational eigenfunctions ψ n 1 n 2 of the ground state.Here, ω c0 is the energy of the vertical transition 0 → c, h (c) and = 0.08 eV are the 2D Hamiltonian and the lifetime broadening of the core-excited state, respectively.Figure 1(a) depicts the localization of the nuclear wave packet in each core-excited state along the reaction path Q 1 which is directed into a valley of the corresponding PES.For the |O1s −1 4a 1  1 dissociative state, the coordinate Q 1 is oriented along one OH bond (Q 1 = R 1 or R 2 ) and for the bound |O1s −1 2b 1 2 state, it is located between the OH bonds (i.e., along the symmetric stretching coordinate Q 1 = Q s ).The wave packet evolving along Q 1 is narrow in the direction Q 2 orthogonal to Q 1 .We describe the strong confinement of in the Q 2 direction by the sharp function (Q 2 ) (normalized to one) using the following factorization: where To connect the projection of this narrow wave packet (7) onto the ground-state PES with the potential V (Q 1 ,Q 2e ) along the confinement coordinate Q 1 we study the relation between the center of gravity cg n (5) and the nth group to the eigenvalue n of the 1D Hamiltonian h 1 along Q 1 : of the 1D Hamiltonian h 1 is not important because we define the potential with respect to the bottom of the well.Clearly, the spectrum of the 1D Hamiltonian n depends on the statesensitive propagation direction of the wave packet φ(Q 1 ), in full agreement with experiment and simulations.Let us write the normalized wave packet φ(Q 1 ) in terms of eigenfunctions ψ n (Q 1 ): One should notice that the overlap between |ψ m and |ψ n 1 n 2 which belong to different groups is negligible due to the small spacing δ n 1 n 2 between the levels inside of the same group (n 1 + n 2 = n) in comparison with the spacing n − m between the levels from different groups (n = m) [11,17] (see Appendix): This together with the identity ψ m |ψ n = 0 allows us to write Eq. ( 5) as follows: Thus Eqs. ( 5) and ( 11) bring about a central result of our article: It is striking that the center of gravity of the nth RIXS peak is nothing else than the eigenvalue n of a 1D Hamiltonian Hence, the set of centers of gravity of the peaks constitutes a pseudospectrum from which we can reconstruct a 1D potential along Q 1 of the multidimensional PES, either by employing an analytical model or a model-free numerical procedure.The pseudospectra, related to the selected cuts, are shown in Fig. 2 for the theoretical RIXS spectra computed at the |O1s −1 4a 1 1 and |O1s −1 2b 1 2 resonances.Figure 2 shows the sensitivity of the pseudospectrum on the direction of the 1D cut.

VI. EXPERIMENTALLY RECONSTRUCTED POTENTIALS
Now, we are in position to validate our concept by extracting the potential cuts from the experimental RIXS spectra of H 2 O (Fig. 3).To reconstruct the 1D cuts we have fitted the experimental pseudospectrum ν to the spectrum of the Morse potential [27] V by varying the parameters ω e and ω e x e .Here the dissociation energy D = ω 2 e /(4ω e x e ) and α = √ 2μω e x e relate to the fitted parameters.
We start the PES reconstructions, from experimental RIXS data, by studying the scattering through the dissociative |O1s −1 4a 1 1 core-excited state.The molecule undergoes an ultrafast dissociation along the OH bonds (along R 1 and R 2 , see Fig. 1), which occurs during the O 1s core-hole lifetime of 1/2 = 4.1 fs.This gives rise to a long stretching vibrational progression, which allows to reconstruct the 1D potential in the direction of the OH bond The RIXS spectrum at the |O1s −1 4a 1  1 resonance [see Fig. 3(a)] exhibits exclusively stretching excitations.Naturally, this progression is not contaminated by the vibrations of the fragment of dissociation OH, as the signal of this species forms the so-called atomic peak [8,28], which is well separated from the main progression in RIXS.The fit of the pseudospectrum n = cg n from the vibrationally resolved RIXS spectrum by the eigenstates of the Morse potential V M (Q − Q e ) with R e = 1.81 a.u.provides us the vibrational FIG. 3. Reconstruction of the 1D cut of the PES along the R 1 , Q s (between the OH bonds) and Q b coordinate from the experimental |O1s −1 4a 1 1 , |O1s −1 2b 1 2 , and |O1s −1 2b 1 1 RIXS spectra due to strong wave-packet confinement.(Left) The centers of gravity of each vibrational peak form the pseudospectrum (colored horizontal lines) of the ground-state 1D potential along the coordinates R 1 , Q s , and Q b selected by the nuclear wave packets highly confined in the valleys of the core-excited states.The fit of the pseudospectrum to the Morse potential V M [27] defines its parameters ω e = (475.4± 1.1) meV and ω e x e = (9.99 ± 0.14) meV for |O1s −1 4a 1  1 and ω e = (457.65 ± 0.70) meV and ω e x e = (4.76± 0.13) meV for |O1s −1 2b 1 2 .The fit of the bending potential eigenstates yields ω e = (201 ± 2) meV and ω e x e = (1.9 ± 0.4) meV for the model V b .Used effective masses are given in the text.(Right) The reconstructed potentials (colored) are in good agreement with the ab initio potentials (black) in the energy range of available vibrational states.
frequency ω e = (475.4± 1.1) meV and the anharmonicity constant ω e x e = (9.99 ± 0.14) meV.Together with the value of the reduced mass μ 1 = m H m O /(m H + m O ) = 0.94 a.m.u., this yields the following parameters of the Morse potential: α = 1.122 ± 0.008 a.u., D = 5.66 ± 0.09 eV.In spite of our experimental data not providing n in the vicinity of the dissociation limit, the obtained dissociation energy is rather close to the single O-H bond dissociation energy of ∼5.6 eV in a gas-phase water molecule [29].The reconstructed potential is compared to the calculated 1D potential along the OH bond in Fig. 3(a).Both potentials are almost identical in the range where vibrational excitations are detected.
Let us tune the photon energy in resonance with the |O1s −1 2b 1 2 core-excited state, where the wave packet is confined within the narrow potential well (Fig. 1) between the bonds.Contrary to the previous case, we are able now to get a cut of the ground-state PES along symmetric bond elongations Q 1 = Q s (see Fig. 1).The stretching pseudospectrum n = cg n extracted from the |O1s −1 2b 1  2 RIXS spectrum fitted by the eigenstates of the Morse potential V M gives values of the vibrational frequency ω e = (457.65 ± 0.70) meV and the anharmonicity constant ω e x e = (4.76± 0.13) meV.Taking the reduced mass for the symmetric stretching bond elongation μ s = 1.044 a.m.u.into account we get α = 0.82 ± 0.02 a.u., D = 11.0 ± 0.3 eV as parameters of the Morse potential V M .The deviation of the obtained value of D from the thermochemical dissociation energy of 9.5 eV [31] is caused by the fact that the vibrational progression is not long enough for the studied state to accurately predict the dissociation limit.Comparing the reconstructed potential to the ab initio cut along Q s in Fig. 3(b), it becomes obvious that there is a nearly perfect overlap between experiment and the theoretically predicted potential in the probed range.
Lastly, we consider the scattering through the |O1s −1 2b 1 1 resonance [see Fig. 3(c)].This spectrum exhibits only bending excitations because the 2D stretching potentials of the |O1s −1 2b 1 1 and ground states are the same [11].Thus, the nuclear wave packet in the |O1s −1 2b 1 1 core-excited state propagates only along the bending coordinate Q b [Fig. 1 The morse potentials and U (Q − Q e ) are matched at the points Q e ± , where is a fitting parameter.The parameters ω e = (201 ± 2) meV, ω e x e = (1.9 ± 0.4) meV, and > Q e were determined by a fit to the spectrum of the potential V b to the experimental data, with the mass of the bending mode μ = μ b = 1.083 a.m.u.The reconstructed potential along the bending coordinate Q b is compared to the ab initio simulation in Fig. 3(c).The limited number of vibrational states in the experimental spectrum does not allow to define precisely the height U max of the barrier U (Q − Q e ).However, the least-squares method employed yields a confidence interval of 1.09 eV < U max < 1.70 eV [see Fig. 3(c)] for the linearity barrier.This confidence interval is consistent with the ab initio values given by our RASPT2 calculations (U max = 1.42 eV) and the coupled cluster value from Ref. [30] (U max = 1.37 eV).

VII. CONCLUSIONS
A rigorous calculation of multidimensional potentialenergy surfaces is a formidable computational task, especially far away from equilibrium where accurate, but computationally costly, multiconfigurational techniques are required.Hence, there is a vital need to reconstruct the potentials directly from the experiment.In this study we validated a method to experimentally extract one-dimensional cuts through the multidimensional ground-state PES of molecular systems using vibrationally resolved RIXS spectra of the water molecule.We utilized the diverse nature of three different O 1s core-excited PES to prepare highly directional nuclear wave packets which were projected onto the ground state in the RIXS process.The physical reason for the reduction of dimensionality, and for the related selection of the directions of the one-dimensional cut, is the high confinement of the wave packet along the valleys of the PES oriented differently for each core-excited state.The quantum-mechanical justification for the extractions of the selected one-dimensional potential cuts was given by relating the pseudospectrum, defined as the centers of gravity of clustered vibrational peaks measured in RIXS spectra, to the eigenstates of the one-dimensional Hamiltonian along the direction of propagation of the core-excited wave packet.
Core-excited states of different elemental sites (C,O,N, etc.) have distinct PESs.These manifolds of intermediate-state PES topologies significantly enlarge the number of directions for 1D cuts of PES which are accessible through RIXS.As shown by many experimental studies of polyatomic systems [5,13,32,33], only a few vibrational modes related to the core-excited atom are active.This state-and site-specific selection of only a few excitation-center-related vibrational modes, together with direct access to high vibrational states in a single shot, makes RIXS a powerful method, yielding access to cuts through PESs in molecules over highly distorted geometries, which complements other vibrational probes.

FIG. 1 .
FIG. 1. Localized nuclear wave packets and 1D projections.Squared core-excited-state nuclear wave packets | | 2 of gas-phase water versus (a) OH bond lengths R 1 and R 2 and (b) bending normal coordinate Q b .exhibits a strong localization along the bonds and between the bonds for the |O1s −1 4a 1 1 and |O1s −1 2b 1 2 core-excited states, respectively, and is altered exclusively along Q b for the |O1s −1 2b 1 1 excited state.The core-excited-state potential-energy surfaces V CE drive directional wave-packet transformations.The projection ψ n 1 n 2 | of the core-excited wave packet onto the ground state allows to reconstruct 1D cuts of the ground-state potential V GS along distinct directions.

FIG. 2 .
FIG. 2. From eigenstates of a multidimensional Hamiltonian to a one-dimensional pseudospectrum.(a)White curves show the crossings of the stretching isoenergetic surfaces n 1 n 2 with the ground-state PES V (R 1 ,R 2 ) = n 1 n 2 .Circles along the coordinates R 1 (green) and Q s (blue) are the centers of gravity cg n of the nth group of the |O1s −1 4a 1 1 and |O1s −1 2b 1 2 .The curves show the theoretical RIXS profile computed at the (a) |O1s −1 4a 1 1 and (b) |O1s −1 2b 1 2 resonances.The bars above show the pseudospectra n = cg n which can be extracted along different coordinates from the |O1s −1 4a 1 1 (b) and |O1s −1 2b 1 2 (c) RIXS due to different Franck-Condon factors (shown by black vertical lines).They allow for the reconstruction of the potential along the OH bonds R 1 and along the symmetric stretching direction Q s .
(b)].The projection of trapped in the valley of the bending potential onto the ground state allows us to recover the 1D cut of the ground-state PES along Q b .The bending potential of H 2 O has two minima (Q e1 = 0 and Q e2 = 3.891 a.u.), symmetric around the angle θ = ∠(H OH ) = 180 • , which corresponds to the top of the barrier at Q e = 1.632 a.u.Therefore, we use a piecewise defined potential function V b formed by two nonoverlapping Morse potentials