Ionization of pyridine: Interplay of orbital relaxation and electron correlation.

The valence shell ionization spectrum of pyridine was studied using the third-order algebraic-diagrammatic construction approximation scheme for the one-particle Green's function and the outer-valence Green's function method. The results were used to interpret angle resolved photoelectron spectra recorded with synchrotron radiation in the photon energy range of 17-120 eV. The lowest four states of the pyridine radical cation, namely, 2A2(1a2-1), 2A1(7a1-1), 2B1(2b1-1), and 2B2(5b2-1), were studied in detail using various high-level electronic structure calculation methods. The vertical ionization energies were established using the equation-of-motion coupled-cluster approach with single, double, and triple excitations (EOM-IP-CCSDT) and the complete basis set extrapolation technique. Further interpretation of the electronic structure results was accomplished using Dyson orbitals, electron density difference plots, and a second-order perturbation theory treatment for the relaxation energy. Strong orbital relaxation and electron correlation effects were shown to accompany ionization of the 7a1 orbital, which formally represents the nonbonding σ-type nitrogen lone-pair (nσ) orbital. The theoretical work establishes the important roles of the π-system (π-π* excitations) in the screening of the nσ-hole and of the relaxation of the molecular orbitals in the formation of the 7a1(nσ)-1 state. Equilibrium geometric parameters were computed using the MP2 (second-order Møller-Plesset perturbation theory) and CCSD methods, and the harmonic vibrational frequencies were obtained at the MP2 level of theory for the lowest three cation states. The results were used to estimate the adiabatic 0-0 ionization energies, which were then compared to the available experimental and theoretical data. Photoelectron anisotropy parameters and photoionization partial cross sections, derived from the experimental spectra, were compared to predictions obtained with the continuum multiple scattering approach.

relaxation of the molecular orbitals in the formation of 7a 1 (n) 1 state. Equilibrium geometric parameters were computed using the MP2 (second-order Møller-Plesset perturbation theory) and CCSD (coupled-cluster singles and doubles) methods, and the harmonic vibrational frequencies were obtained at the MP2 level of theory for the lowest three cation states. The results were used to estimate the adiabatic 0-0 ionization energies, which were then compared to the available experimental and theoretical data. Photoelectron anisotropy parameters and photoionization partial cross-sections, derived from the experimental spectra, were compared to predictions obtained with the continuum multiple scattering approach.

I. INTRODUCTION
Pyridine ( Figure 1) is an important heterocyclic molecule, and is closely related to benzene and more complex six-membered nitrogen-containing aromatic molecules. Such molecules occur as structural units in many biological compounds. Hence, an understanding of the electronic structure of pyridine is essential to many fields of organic chemistry including biochemistry, medicinal and pharmaceutical chemistry. 1,2 Photoelectron spectroscopy, especially when carried out with a continuously tunable photon source, provides an ideal means of exploring molecular electronic structure. The results obtained from such experiments can be interpreted with the help of quantum chemical calculations. The latter can be performed using various approximate computational schemes ranging from Hartree-Fock (HF) theory (Koopmans' theorem) to highly sophisticated methods aiming to eliminate shortcomings of the one-electron approximation. 3 The improvements with respect to the Koopmans level of theory are usually discussed in terms of two effects, namely orbital relaxation and electron correlation. 4 Whereas orbital relaxation (i.e. the response of the electronic structure to a hole created in the occupied orbitals) can be accounted for using HF calculations performed separately for the initial and final states [the so-called delta selfconsistent field (SCF) approach], electron correlation can be treated only through methods going beyond the HF level of theory.
Reliable results for ionization potentials (IPs) can also be obtained within the framework of the equation-of-motion (EOM) coupled-cluster (CC) theory (EOM-IP-CC) [19][20][21] at the level of singles and doubles (CCSD) and higher CC models, 22 or by using the theoretically equivalent symmetry-adapted cluster configuration interaction (SAC-CI) method. 23 Accurate predictions of ionization energies for the lowest cationic state in each spatial symmetry are possible when separate CC calculations for the neutral and cationic species are performed (CC approach).
Similar effects concerning nearly degenerate energy levels, due to the ionization of nonbonding n-type lone-pairs (LPs) and -orbitals, has been observed in other heteroaromatic molecules, such as adenine, [49][50][51] pyrimidine 52 and purine, 52 where the LPs belong to the nitrogen atoms. In thymine, 50,51,53 uracil, and the uracil derivatives, [54][55][56] the LPs involved belong to the oxygen atoms outside the aromatic ring; and in guanine 51,57,58 the LPs have a mixed nitrogen-oxygen character.
Another important issue related to the theoretical treatment of the nearly-degenerate n- pairs is that the n-and -states are initially always strongly separated at the HF (Koopmans' theorem) level of theory, but merge together when a more accurate treatment is employed. The shifts in the HF ionization energies are non-uniform, and are much larger for the n-levels than for the -levels. This effect is more pronounced in the propagator results 35,47,48,50,[53][54][55][56][57][58][59] than in those employing the EOM-IP-CC approach. 51 Previously, the ionization of pyridine has been studied theoretically by Suzuki and Suzuki, 41 using the continuum multiple scattering X (CMS-X) method, by Wan et al., 42 and Kitao and Nakatsuji, 43 using the SAC-CI method, and by Plashkevych et al. 44 using Kohn-Sham density functional and transition potential methodology. Yang et al. 45 used the direct static exchange approach, which takes cross-section effects into account, to investigate the photoelectron spectra of pyridine and some of its derivatives. Lorentzon et al. 46 computed the lowest ionization potentials using second-order complete-active space perturbation theory (CASPT2). Walker et al. 40 studied a large number of cationic states spanning the entire outervalence energy domain using multi-reference configuration interaction methods with singly and doubly excited configurations (MRD-CI). The ionization spectrum of pyridine has also been treated using many-body Green's function techniques. 35,47,48 A comprehensive review of the corresponding experimental work, carried out prior to 1988, has been provided by Innes et al. 24 Many of these early studies focused on identifying the molecular orbitals giving rise to the first two bands in the photoelectron spectrum. The outermost band in the HeI excited photoelectron spectrum, [25][26][27][28][29][30] observed at binding energies between 9.2 and 10.2 eV, is predicted to correspond to ionization from the 7a 1 (σ N LP ) and 1a 2 () orbitals. This band is partially overlapped by the second band, centred around 10.5 eV and due to the 2b 1 () orbital ionization. Band shape considerations led Gleiter et al. 26 to associate the 7a 1 and 1a 2 orbitals with the highest occupied molecular orbital (HOMO) and HOMO-1, respectively, although King et al., 27 based upon fluoro-substituent effects, preferred the reverse ordering.
Another experimental approach which is often useful in assigning a particular molecular orbital to a specific photoelectron band involves measuring the photoelectron angular distributions. This technique was used in the HeI excited photoelectron study performed by Utsunomiya et al. 29 to determine the anisotropy parameters, as a function of binding energy, of the outermost bands of pyridine and dimethylpyridine. For the dimethyl substituted molecule, the molecular orbital sequence is well established. A comparison between the anisotropy parameters for dimethylpyridine and those for pyridine resulted in Utsunomiya et al. assigning the lowest ionization in pyridine to the 7a 1 orbital.

Photoelectron angular distribution experiments have also been performed by
Piancastelli et al. 31 employing synchrotron radiation. Their measurements showed that the anisotropy parameter for the high binding energy region of the first band increased more rapidly as the photon energy increased from 13 to 27 eV than did the anisotropy parameter for the low binding energy region. Since a rapid increase in the value of the anisotropy parameter is typical for electron ejection from a -orbital, Piancastelli et al. 31 associated the low binding energy region of the first band predominantly to contributions from the 7a 1 (σ N LP ) orbital. Similar conclusions were reached by Berg et al. 32 using multiphoton ionization.
Subsequent experimental studies include photoelectron spectra recorded with HeI 33,34 or synchrotron 35,36 radiation, a resonance-enhanced multiphoton ionization study, 37 and mass analyzed threshold ionization (MATI) investigations. 38, 39 The emphasis of these recent studies lay in obtaining an improved understanding of the spectroscopic properties of the pyridine cation. In contrast, the valence shell photoionization dynamics have not been investigated in a detailed manner, either experimentally or theoretically.
The results from earlier calculations have allowed many of the bands observed in the photoelectron spectrum to be assigned but, as already mentioned, inconsistencies remain with respect to the position of the 2 A 1 (n 1 ) state and of some of the other states. The difficulty concerning the 2 A 1 (n 1 ) state is related to a more general issue regarding the ionization of n-orbitals in heteroatomic molecules. The present work addresses this issue by analyzing the lowest and -type cationic states of pyridine in terms of orbital relaxation and electron correlation effects. We use the ADC(3), OVGF and EOM-IP-CC methods and present an analysis of relaxation energies based on second-order perturbation theory, together with electron density difference plots, and Dyson molecular orbitals. We also measure valence shell photoelectron angular distributions and photoionization partial cross-sections, and compare these experimental results to theoretical predictions obtained with the CMS-X approach, to help characterize the photoionization dynamics.

II. THEORY AND COMPUTATIONAL DETAILS OF ELECTRONIC STRUCTURE CALCULATIONS
The computation of the energies and photoelectron intensities of the outer-and innervalence vertical ionization transitions of pyridine was based on the ab initio Green's function (GF) electronic structure methods as used in our previous studies of halogenated benzenes 11,12 and other heterocyclic molecules. 50,[53][54][55]57 More specifically, the ADC(3) scheme for the one-particle Green's function [5][6][7][8] was employed. For the outer valence orbitals the less rigorous but more economic OVGF method 7,14 was also used. The results of the calculations were analyzed with the aid of Dyson molecular orbitals for the lowest four ionization transitions.
The electron density difference maps for HF and coupled-cluster doubles (CCD) densities were plotted, and the second-order perturbation-theoretical treatment of the relaxation energy was employed to obtain a better insight into the electron correlation and orbital relaxation effects in the ionization of these orbitals.
A. Third-order algebraic-diagrammatic construction approach to the oneparticle Green's function In the ADC(3) scheme for the one-particle Green's function, 5-8 the vertical ionization energies and the corresponding transition probabilities are obtained as the solutions of the Hermitian eigenvalue problem: where the matrix B is defined according to: Here ε is the diagonal matrix of the HF orbital energies, and ( )  Σ is the constant part of the self-energy. Both quantities are defined with respect to the space of one-hole (1h) and oneparticle (1p) configurations with respect to the HF ground state. The matrices ( )   K C and  U are referred to as matrices of effective interactions and effective coupling amplitudes, respectively. Their definition involves configuration spaces of two hole-one particle (2h-1p) and two particle-one hole (2p-1h) configurations, indicated by "minus" and "plus" superscripts, respectively. The elements of the ( ) The spectroscopic factors ( ) n p x (also known as Feynman-Dyson amplitudes) allow one to evaluate the "pole strength" P n , which is used as an approximate measure of the relative spectral intensity: where the summation is performed over the whole range of molecular orbitals (MOs).
In the ADC(3) method, both n e and ( ) n p x are treated consistently through third order of PT with respect to the main photoelectron transitions (producing cationic states of 1h-type).
The photoelectron satellite states characterized as 2h-1p excitations are treated consistently only through first order, so that a less accurate description is afforded in spectral regions dominated by such transitions. Nevertheless, the ADC(3) method allows a fairly reliable description of cationic states characterized as strong 1h/2h-1p mixtures. Such mixtures are often encountered at higher ionization energies where a "break-down of the one-electron picture of ionization" may occur. 4 This capability is not only important for an adequate description of the inner-valence region of the ionization spectra, but also for the treatment of heterocyclic molecules (which includes pyridine) where an early onset of the "selective" break-down phenomenon, characterized by the strong redistribution of intensity from the inner -type MOs to various satellites, has been observed. 61,62 In general, the ADC(3) method takes into account all physical effects, such as electron correlation in the initial and final states, and orbital relaxation, relevant to ionization. The computational procedure of the ADC(3) scheme is size-consistent, 63 and can be implemented as an n 5 scaling of the computational effort, where n is the number of molecular orbitals (disregarding here the improved fourth-order treatment of ( )  Σ ). This represents one of the best trade-offs between computational effort and theoretical accuracy available today.

B. Dyson orbitals
The where the integration is performed over the space-spin coordinates i x of (N1) electrons. Eq.
(5) can be rewritten using the second-quantization formalism as follows:  10 The following important relation can be obtained for DOs: This implies that the norm of n  is given by the pole strength n P of the corresponding transition. According to Eq. The Dyson orbitals can be used for studies of various ionization related phenomena, [67][68][69][70][71] as well as for the visualization of the results of the Green's function calculations. 49,53 In the present study we employ Dyson orbitals to analyze the ADC(3) results for pyridine.

C. The electron density and electron density differences
Another option to analyze the changes in the electronic structure upon ionization is to consider the many-electron density differences. The initial N-electron ground-state density where 0 N  as earlier is the ground-state wavefunction; the integration is performed over the space coordinates i r of (N1) electrons and the summation is performed over all spin coordinates s i . Analogously, the final-state (N1)-electronic density can be written as: is the wavefunction for the cationic state under consideration and the integration is performed over the coordinates of (N2) electrons.
The density difference function shows the changes in the electron density upon ionization. In contrast to the Dyson orbitals, is not affected by the limitations with respect to 2h-1p admixtures, and correlated wave functions of arbitrary complexity can be studied. In the present work, we inspect to qualitatively investigate orbital relaxation and electron correlation effects in pyridine, using densities computed at the CCD level of theory.
The electron density differences provide a possibility of comparing different theoretical approximations. By comparing the cationic electron densities computed using the HF and SCF approaches the changes due to the orbital relaxation, -1(rel) ( ) N n   r , can be isolated and analyzed: Similarly, when the SCF electron density for a cationic state is compared to the density computed using the CCD approach based on relaxed SCF orbitals (within the unrestricted HF framework), a density difference function of the form allows for an insight into the role of the electron correlation effects.
To display ( )   r , which depends on three spatial coordinates, we use two-dimensional contour plots (where two coordinates are varied and one coordinate remains fixed). As sections we chose the plane of the pyridine ring and the parallel plane 0.5 Å above the molecular plane.

D. Second-order perturbation theory for the relaxation energy
As mentioned above, the SCF approach allows one to fully recover the relaxation energy. The SCF results can, in turn, be analyzed by perturbation theory (see Refs. [66,73] and references cited therein), providing useful insight into the nature of the relaxation processes.
Here and further on O(3) denotes the remainder of perturbation series beyond second order.
Here, i and a denote occupied and unoccupied orbitals, respectively, in the HF ground-state.
The Coulomb two-electron matrix elements (pq|rs) are given in the "1122" notation, and p  denote the HF orbital energies. Here and further on in this section the expressions are given in the spin-free form where the summation over spin variables has been performed so that only spatial orbitals come into play.
For further use it can be convenient to rewrite the relaxation energy as follows: In Eq. (16) each term ( , ) S k ia consists of the two parts: Here we also note that the matrix element ( | ) kk ai can be considered as the matrix element ˆk i J a of the familiar Coulomb operator ˆk J for orbital k.

E. Details of the computations
The ADC(3) calculations were performed using the original code 74 linked to the Gamess ab initio program package. 75 The cc-pVDZ basis set 76  The ground-state geometrical parameters of pyridine were obtained through a full geometry optimization using MP2 theory together with the cc-pVTZ basis sets. 76 The MP2 calculations were carried out using the Gaussian package of programs. 77 Several basis sets were used in the OVGF calculations. 7,16 Besides the cc-pVDZ basis, allowing for a comparison of the OVGF and ADC(3) results, the hierarchy of cc-pVnZ (n = D, T, Q, 5) and aug-cc-pVnZ (n = D, T, Q) correlation consistent basis sets 76 with spherical representation of d-functions were also used.
The frozen-core approximation was adopted in the ADC(3) and OVGF calculations, that is, the carbon and nitrogen K-shell orbitals were not correlated. This frozen-core model has been adopted in the notation of the molecular orbitals throughout this paper.
The results of the OVGF calculations allow the convergence of the ionization energies with respect to systematic improvements in the basis set to be studied and to extrapolate to the CBS limit. Here we use the three-parameter exponential formula: 78-80 where n is the cardinal number of the basis set (2, 3, 4, 5 for D, T, Q, and 5 sets, respectively), CBS E is the estimated ionization energy for the CBS limit (as n   ), and B,  are further parameters found by fitting Eq. (20) to the computed energies ( ) E n . This type of extrapolation is known to provide sufficiently reliable results, [78][79][80] although some other techniques can be more advantageous for extrapolations of the ionization energies. 69 The outer-valence vertical ionization energies were also computed using the EOM-IP-CCSD method, [19][20][21][22] as implemented in the Q-Chem program package. 81 The aug-cc-pVTZ basis set 76 was used in these calculations.
The restricted Hartree-Fock (RHF) and unrestricted Hartree-Fock (UHF) calculations were performed to obtain HF, RHF/SCF, and UHF/SCF ionization energies. The calculations were carried out using the Gaussian, 77 Q-Chem, 81 and Gamess 75 program packages. In the SCF calculations, in addition to the 1 A 1 ground-state of neutral pyridine, the ) states of the pyridine cation-radical were also considered.
In the ""-approach, the total energies of the initial and final states are computed, and the transition energies are then given as total energy differences. In the present work the approach to ionization energies was applied at the MP2 (second-order Moller-Plesset PT), MP3, and MP4 levels, as well as by using coupled cluster methods at the CCD In order to obtain the adiabatic and 0-0 transition energies, the geometry optimization for the 2 A 1 (7a 1 1 ), 2 A 2 (1a 2 1 ), 2 B 1 (2b 1 1 ), and 2 B 2 (5b 2 1 ) cationic states was performed at the MP2 and CCSD levels of theory using the cc-pVTZ basis set. The geometry optimization for the 2 B 2 (5b 2 1 ) state carried out using the UHF-based MP2 method failed. Regardless of the initial guess, large gradients were obtained which caused a reduction of the molecular symmetry to the C 1 point group. All cationic states then fall into the same irreducible representation, and the variational UHF method converges to the lowest state 2 A 2 (1a 2 1 ). The optimized geometric parameters for the 2 A 2 (1a 2 1 ) state were therefore obtained. The equilibrium electronic ground-state geometry of neutral pyridine was also computed using the same methods for a reference. For all states where MP2-optimized geometric parameters could be obtained, the harmonic vibrational frequencies were evaluated at the MP2 level of theory. The calculations were carried out using the Gaussian 77 program within the UHF and frozen-core frameworks.
The second-order perturbation theory for the relaxation energy, presented in Sec. IID, was implemented as a supplement to our ADC software. 74 The calculations were performed using the compact 6-31G* basis set 84 to simplify the analysis of the results.
The Gaussian Cube utility 77 was used to generate the total electron densities ( )  r and the electron density differences ( ) r function (Eq. (10)) was evaluated at the CCD level of theory within the UHF and frozen-core frameworks. The -1(r el) N n   function (Eq. (11)) was generated from the RHF/SCF and HF densities. The latter were obtained as follows: the neutral ground-state HF orbitals were given as the initial guess for the cationic-state RHF calculations and the SCF procedure was terminated before the onset of the iterations. The was used to plot the Dyson orbitals computed by our one-particle Green's function program 74 at the ADC(3)/cc-pVDZ level of theory.

F. CMS-Xα calculations of photoionization cross-sections and photoelectron angular distributions
Photoionization dynamics calculations were performed using the CMS-Xapproach, following procedures similar to those described previously. 87,88 Briefly, a model potential is constructed using overlapping, spherical atomic volumes, placed at centres determined from the MP2/cc-pVTZ optimized geometry, with atomic radii selected and scaled (0.85) as previously described. 88 We then iterate to achieve a self-consistent ground state potential in which the exchange contribution to an effective one-electron potential is represented using the Xlocal density approximation. A final state cation potential having the correct asymptotic Coulombic form was subsequently generated from this ground state neutral potential. Electron wavefunctions, expanded in a symmetry adapted basis of spherical harmonic functions, and with radial terms obtained by direct numerical integration, are then obtained in these potentials and used to construct the electric dipole photoionization matrix elements in an independent electron, frozen core approximation. Finally, photoionization cross-sections and photoelectron anisotropy parameters were computed from these matrix elements, and thus represent ionization of a fixed geometry, non-rotating, non-vibrating molecule.
The spherical harmonic basis expansions for the initial neutral state were truncated at values, l max , of 5, 2 and 1 in, respectively, the outer molecule region, the non-hydrogenic atomic regions, and the hydrogen atoms regions. For the final continuum state, the outgoing electron can be scattered into higher l-waves by the anisotropic ion potential, and so the corresponding truncation limits were extended to l max = 9, 4, and 3.

III. EXPERIMENTAL APPARATUS AND PROCEDURE
Photoelectron spectra of pyridine were recorded using a rotatable hemispherical electron energy analyzer and synchrotron radiation emitted from the Daresbury Laboratory storage ring. 89 Detailed descriptions of the monochromator 90 and the experimental procedure have been reported. 91 The photoionization differential cross-section in the electric dipole approximation assuming randomly oriented targets and electron analysis in a plane perpendicular to the photon propagation direction, can be expressed in the form where σ total is the angle-integrated cross-section, β is the photoelectron anisotropy parameter, θ is the photoelectron ejection angle relative to the major polarization axis and P is the degree of linear polarization of the incident radiation. At each photon energy, photoelectron spectra were recorded at  = 0 and  = 90, thus allowing the anisotropy parameter to be determined once the polarization had been deduced. The degree of polarization was determined by recording Ar 3p and He 1s photoelectron spectra as a function of photon energy, and using the well-established β-parameters for these gases. 91,92 The spectra were analyzed by dividing the binding energy range into the regions specified in Table I to obtain photoelectron anisotropy parameters and β-independent branching ratios. The branching ratio for a specific region is defined as the intensity in that particular region divided by the sum of the intensity in all the energetically accessible regions.
Since the binding energies for some of the orbitals are very similar, the corresponding photoelectron bands overlap to some extent, and for these bands a complete separation between the ionic states is not possible.
The branching ratios (not shown) for the outer valence shell were used to derive absolute photoionization partial cross-sections (regions 1 -8, Table I) as follows. First, the absolute total photoionization cross-section was obtained as the product of the absolute photoabsorption cross-section 93 and the photoionization quantum efficiency. 93 Secondly, the absolute photoionization partial cross-section for a particular region (Table I) was then determined by multiplying the appropriate branching ratio with the absolute total photoionization cross-section.

IV. RESULTS AND DISCUSSION A. Pyridine molecular orbitals
According to our HF calculations, the ground-state valence-shell electron configuration of pyridine can be written using C 2v point group notation as where the core orbitals are excluded from the orbital numbering scheme. The character of the outer-valence orbitals can be understood from the Mulliken atomic populations given in Table II.
The MOs can be assigned as and -type, as is usual for aromatic molecules. The aromatic system comprises three -orbitals (1b 1 , 2b 1 , 1a 2 ). A characteristic feature of the pyridine molecule is the non-bonding -type lone-pair orbital of the nitrogen atom (7a 1 ). This orbital appears in the spectrum at a low binding energy, although the exact location remains uncertain. According to the Mulliken populations (Table II), the nσ N LP orbital shows a substantial localization on nitrogen, although some of the electron density is delocalized over the hydrogen atoms. The other orbitals forming the outer-and inner-valence shells of pyridine are essentially -type bonding orbitals with significant delocalization over the ring and hydrogen atoms.

B. Vertical ionization energies
The energies and intensities of the vertical transitions belonging to the outer-valence region of the ionization spectrum, computed using various methods, are listed in Table III. The theoretical results are obtained using the HF, OVGF, ADC(3) and EOM-IP-CCSD methods in combination with the cc-pVDZ and aug-cc-pVTZ basis sets. Experimental values are also given for comparison.
The basis set effect can be studied in more detail using the data given in Table IV, comparing the OVGF results for the aug-cc-pVnZ and cc-pVmZ series up to n = Q and m = 5.
Since the accuracy of the OVGF results above 16 eV is not sufficient for a reliable extrapolation, due to increased configuration interaction effects in this energy region, the data for the 5a 1 , 3b 2 and 4a 1 orbitals were not included. Beyond the cc-pVTZ level, the results are nearly converged, so that the cc-pVTZ and aug-cc-pVTZ sets can be used for a stringent comparison with both experiment and between different theoretical schemes. The cc-pVDZ basis set, on the other hand, is more appropriate for a qualitative description of the spectrum yielding ionization energies typically 0.25 eV below the basis set limit. Overall, the one-particle picture of ionization 4 holds to a good approximation over the entire outer-valence region.
In general, the present ADC (3), OVGF, and EOM-IP-CCSD results are fairly consistent with the experimental data. Disparities, such as those for ionization of the 3b 2 and 4a 1 orbitals, can most likely be explained by the possibility that the measured peak positions do not strictly match the vertical ionization energies. The two quantities can deviate from each other due to extended vibrational progressions and overlapping spectral structures. As already mentioned, the first band in the photoelectron spectrum encompasses two closely spaced transitions due to the 1a 2 (π) and 7a 1 (σ N LP ) orbitals. In view of the error margins in the computational results (Table III), the correct energetic ordering of these states becomes an issue. Whereas the ADC(3) and OVGF methods predict the state ordering to be 1a 2 (π) 1 followed by 7a 1 (σ N LP ) 1 at slightly higher energy, the EOM-IP-CCSD calculations indicate the reverse order. The lack of agreement amongst the theoretical methods with respect to the low-energy part of the ionization spectrum calls for further attention. To address this issue, in Table V we list the results for the four lowest vertical ionization energies obtained in the present work using different computational approaches, as well as those from previous studies. 35,40,42,46 As can be seen, the difficulty arises mainly from the large variation in the estimates for the nitrogen LP orbital, 7a 1 (σ N LP ), whose ionization energy depends strongly on the method used.
The 7a 1 (σ N LP ) ionization energy, being grossly overestimated at the HF level At the highest level of theory (EOM-IP-CCSDT) employed in the present work, the 7a 1 (σ N LP ) 1 state is predicted to lie vertically below the 1a 2 (π) 1 state by 0.12 eV. The EOM-IP-CCSDT/cc-pVTZ results, owing to the high-level of the CCSDT approximation and the cc-pVTZ basis (Table IV), provide a very accurate description of the electronic structure.
Using the EOM-IP-CCSDT/cc-pVTZ results as benchmarks for the other computational schemes in Table V  In view of the apparent complexity of the lowest cationic states of pyridine, a more detailed analysis seems appropriate. First, one can look at the Dyson orbitals, which represent many-electron effects in orbital form, visualizing the molecular domain left by the emitted electron ( Figure 2). The Dyson orbitals for the 1a 2 (π) 1 and 2b 1 (π) 1 states closely resemble the corresponding canonical HF molecular orbitals, both being of typical -character. The Dyson orbital for the 7a 1 (σ N LP ) 1 state is also rather similar to the corresponding HF MO. Its lone-pair character is clearly reflected in the characteristic density maximum, directed perpendicular to the ring plane, at the nitrogen site. For the 5b 2 (σ) 1 state the Dyson and HF orbitals describe -bonding of the N-C 2 , C 3 -C 4 , and C-H atoms.
More information can be gained from the density difference plots comparing the total correlated electron density in the electronic ground state of neutral pyridine and its cationic states. Since total densities cannot be obtained in the ADC approach, they were evaluated using the CCD method for both the initial and final states ( Figure 2). This should be sufficient for a qualitative analysis. The regions in the density difference plots shown in blue are depleted as a result of electron emission. The shape of these areas agrees with the shape of the Dyson orbitals when the phase structure is disregarded in the latter.
The Dyson orbitals and the density difference plots are equally useful for visualizing regions of decreased electron density. The density difference plots, however, also give an idea of the density redistribution in the cations due to orbital relaxation and electron correlation. In Figure 2, the regions plotted in red are characterized by increased electron population and reflect the rearrangements taking place in the cationic states. As can be seen, the areas of increased density are compactly located in the central part of the molecule, mainly on the bonds forming the six-membered ring. This can be understood in that the remaining density tends to strengthen the bonds in the cation-radicals compensating for the loss of an electron.
The observed rearrangements can also be interpreted as a screening of the positive charge of the hole generated by the removal of an electron. For the -type final states, 2 A 2 (1a 2 1 ) and 2 B 1 (2b 1

1
), the screening effect is observed in the molecular plane, which is the nodal plane of the -orbitals. For states of -symmetry, 2 A 1 (7a 1

1
) and 2 B 2 (5b 2 1 ), in contrast, some increase of electron density can also be seen above (and below) the molecular plane.

D. Orbital relaxation and electron correlation effects
It has yet to be explained why the ground-state HF orbital (Koopmans' theorem) provides such a poor description of the 7a 1 (σ N LP ) 1 state, necessitating a large correction at the higher computational levels. It is instructive to note that the SCF treatment yields quite reasonable estimates for the relative ionization energies (or spacing) associated with the three lowest states, 1a 2 (π) 1 , 7a 1 (σ N LP ) 1 and 2b 1 (π) 1 . The SCF scheme fully accounts for orbital relaxation but disregards electron correlation, which indicates that relaxation is a crucial factor in the description of the 7a 1 (σ N LP ) 1 state. Electron correlation, on the other hand, improves the absolute ionization energies by shifting both the neutral ground state and the cationic states to lower energy. The resulting correction, defined by these two shifts, is state specific.

Orbital relaxation
In order to see how relaxation changes the localization properties of the 7a 1 (σ N LP ) and other molecular orbitals, in Table VI we compare their Mulliken atomic populations in the electronic ground state (closed shells) and in the respective cation radicals (half-occupied shells). Table VI shows that the Mulliken atomic population of the 7a 1 (σ N LP ) MO changes dramatically upon ionization. The nitrogen character increases by about 25%, from 62% in neutral pyridine to 88% in the pyridine cation-radical. The 7a 1 (σ N LP ) MO is significantly more localized on nitrogen in the cationic state than in the (neutral) ground state. This can be seen not only in the decreased contribution of the carbon atoms, but also in a quite substantial (7%) decrease in the contributions of the hydrogen atoms. By contrast, the -orbitals, 1a 2 (π) and 2b 1 (π), hardly change upon ionization, and largely retain their localization properties in the cation-radical. The localization properties of the 5b 2 (σ) orbital change moderately. Here a certain decrease of N-and C 2 -character and the corresponding increase of C 3 -and C 4character occurs.
The changes in the total electronic density distribution caused by relaxation can be visualized by the density difference plots (Figure 3) for the 7a 1 (σ N LP ) and 5b 2 (σ) orbitals.
Similar plots for the 1a 2 (π) and 2b 1 (π) orbitals, together with above-plane density difference plots for all four orbitals, are shown in Figure S1 in Supplementary Material. These plots compare the cationic states in terms of the unrelaxed (neutral ground-state HF) density and the fully relaxed density obtained from the ROHF calculations for the state under consideration.
For the 2 A 1 (7a 1 1 ) state the density difference surface is very steep, especially in the vicinity of the nitrogen atom. This implies that a strong rearrangement of the electron density due to relaxation takes place. When relaxation is taken into account, more electron density is withdrawn from the area surrounding the nitrogen atom, whereas there is an increase in the electron density on the molecular peripheral domains including the hydrogen atoms and C-H bonds. The in-plane and above-plane cross-sections show a similar relaxation pattern, although -as expected for states of -symmetry -the effect is much weaker above the molecular plane. This description is in agreement with the picture based on the Mulliken atomic populations given in Table VI, predicting a more pronounced lone-pair character for the 7a 1 orbital at the level of the SCF treatment.
The density difference surfaces of the -states, 2 A 2 and 2 B 1 , are rather flat ( Figure S1), which means that there is no pronounced relaxation effect. For the -states, the density difference above the molecular plane has to be considered. It shows density redistribution patterns similar to the shapes of the corresponding 1a 2 and 2b 1 orbitals. Interestingly, there is also a notable in-plane relaxation effect leading to quite different and substantially more symmetric density redistribution patterns. In both cases, the in-plane electron density is transferred from the six-membered ring towards the hydrogen atoms.
Strong in-plane relaxation effects can be seen for the 2 B 2 (5b 2 1 ) state ( Figure 3). Here the electron density is transferred from the bonds adjacent to the C 4 atom to the bonds on the opposite side of the molecule adjacent to nitrogen. The in-and above-plane reorganization effects differ somewhat, as can be seen from the respective surfaces ( Figure S1).

Electron correlation
The orbital relaxation effects in the pyridine cation-radicals considered above are characterized by a strong non-local density redistribution, resembling to a certain extent intra-molecular charge transfer. In contrast, when electron correlation is taken into account the resulting changes to the electron distribution are essentially local. This is evident from the characteristic pattern in the total electron density difference maps (Figures 3 and S2 in Supplementary Material) obtained by comparing the relaxed UHF densities for the cationic states with the densities derived from CCD computations using the UHF orbitals. For all four states considered here, one sees that electron correlation introduces local corrections which decrease the density on the chemical bonds and increase the density near atoms. This leads to a more balanced electron density distribution, compared to the case when only orbital relaxation effects are taken into account, and is in agreement with the earlier findings concerning the role of the electron correlation effects. 94,95 Considerable in-plane density redistribution due to electron correlation can be seen in the -states, 2 A 2 and 2 B 1 ( Figure S2). As already mentioned, the correlation induced changes modify the local bonding situation. The above-plane cuts of the density difference maps do not facilitate a simple interpretation, but clearly indicate that for both and -type states correlation related changes in the electron density arise throughout the entire molecular environment, including regions distant from the ring plane.
The electron correlation effects for the ionization energies, as obtained by the CCD approach, are quite considerable, amounting to 1.57, 1.18, 1.30, 0.81 eV for the 7a 1 ( LP ) 1 , 1a 2 () 1 , 2b 1 () 1 , 5b 2 () 1 states, respectively (Figures 3 and S2). These corrections are roughly of the same order of magnitude as the relaxation shifts (2.73, 0.85, 0.96, 0.86 eV, respectively), but have the opposite sign (Figures 3 and S1). Except for the 7a 1 ( LP ) 1 state, this leads to a rather effective compensation of the two effects, resulting in modest overall corrections ( Figure S3 in Supplementary Material). For the 2 A 1 (7a 1 1 ) state, however, the relaxation effect is much more substantial than the correlation contribution, so the total correction is large (1.16 eV). The electron density difference maps ( Figure S3 in Supplementary Material), incorporating both effects, support these findings. One can therefore state that the neutral ground-state HF description of the  LP MO in pyridine -and most likely in other heteroaromatic compounds -is inappropriate for the associated cationic states.

Perturbation treatments
As addressed in Sec. IID, the SCF calculations can be analyzed by perturbation theory.
The results of a PT2 analysis for the four highest occupied orbitals are given in Table VII. The SCF and HF ionization energies are also listed, establishing the full relaxation energy, R E  (SCF), according to Eq. (13). As can be seen, the PT2 approach recovers the relaxation energy surprisingly well. The mean error for the and -type states is only 0.2 and 0.1 eV, respectively, with R E  being overestimated in the former and underestimated in the latter.
The IPs derived from PT2 and SCF are in good agreement as well.
The five largest terms in the PT2 expansions of  Table VIII shows that there are obvious differences between the cationic states of  N LP -, -, and -type. According to our results, the  N LP relaxation is driven mainly by the two-electron integrals of ( N LP  N LP |*)  σ N LP π π* J type describing the coupling of the  N LP charge to the electron density distributed in the  and * orbitals. The large relaxation effect in the 7a 1 ( LP ) 1 state is reflected in the substantial magnitude of that Coulomb integral.
Also, the small -* energy gap plays an important role (Table VIII). This finding implies that the -system plays an active role in the screening of the  N LP -hole and in the relaxation accompanying the ionization of an  N LP electron (Eq. (14)). This is also in agreement with the density difference plots demonstrating a considerable above-plane relaxation component for ionization of the  LP -orbital. Indeed, according to our results, more than 50% of the relaxation energy of the  LP orbital is due to the -system contribution (as indicated by the R E   value in Table VII). In contrast, the -system contributes only 28% to the relaxation energy of the 5b 2 () 1 state (Table VII), which clearly shows here a distinctly different relaxation mechanism. For the -type states, 1a 2 () 1 and 2b 1 () 1 , the main relaxation effect appears to be due to the -system, since the -system supplies, in both cases, only 11% of the relaxation energy (Table VII). Thus, the most important two-electron integrals here are of (|*)  π σ σ* J type (Table VIII).
An important distinction between the relaxation mechanism in the 7a 1 ( N LP ) 1 state and that in the 1a 2 () 1 , 2b 1 () 1 , and 5b 2 () 1 states is the existence of a single dominant term in the PT2 expansion (Table VIII). According to our results the largest S contribution for the 7a 1 ( N LP ) 1 state provides 24% of the total relaxation energy, but only 13, 8, and 10% in the 1a 2 () 1 , 2b 1 () 1 , and 5b 2 () 1 states, respectively. In the latter cases no single major contribution can be identified; the individual contributions to the relaxation energy are relatively small as a consequence of small matrix elements in the nominators and large energy gaps in the denominators.
From the above considerations it is clear that the -system, with the low-lying unoccupied *-MOs having similar localization properties as the occupied -MOs, is essential for an efficient screening of the n-holes (the pairs of orbitals 2b 1 ()-3b 1 (*) and 1a 2 ()-2a 2 (*) fulfill this requirement). On the other hand, the n-orbital is well localized, so that large nσ π π* J integrals are obtained. Large-magnitude π π* k J matrix elements, similar to those discussed here in relation to valence shell ionization, occur in K-shell ionization, where k are highly localized 1s-MOs. Huge relaxation shifts are typical for such core-hole states. 66 The charge screening in the  LP MO by the -electrons is accomplished by -* excitations. The relaxation effects are generally taken into account via electron interaction of 1h-states with 2h-1p-, 3h-2p-and higher excited states.  (22) is readily identified as the Coulomb integral ˆk j J a discussed above in the context of the PT2 relaxation energy analysis.
The first-order interaction of the 1h-and 2h-1p-states is explicitly taken into account in the ADC(3) and EOM-IP-CCSD schemes and contributes to the treatment of the orbital relaxation effects. Indeed, inspection of our ADC(3)/6-31G results for the 7a 1 (σ N LP ) 1 state shows that the total weight of the 2h-1p configurations in the ADC(3) eigenvector amounts to 11%, and the most important of these configurations are:

E. Geometric structure and vibrational analysis
Having discussed the vertical ionization energies of the four lowest cationic states, we now look at the geometric structure and vibrations. The optimized geometrical parameters for the 2 A 1 (7a 1 1 ), 2 A 2 (1a 2

1
), and 2 B 1 (2b 1 1 ) states, obtained at the MP2 and CCSD levels using the cc-pVTZ basis set, are given in Tables IX and X, 96 Consistent results are obtained for the angles, and agree well with the experimental data. 96 The neutral ground-state frequencies calculated in this work using the MP2 method agree very well with the experimental data of DiLella et al. 97 The calculated normal modes are consistent with their classical definitions given by Langseth and Lord 98 and by Zerbi et al. 99 The original notations of Langseth and Lord were therefore adopted, while some insight into the nature of the vibrations can be gained from the qualitative descriptions introduced by Kumar et al. 100 also given in Table XI.
The vibrational frequencies of the cationic states are less well studied but Lee et al. 39 have calculated B3LYP/cc-pVTZ frequencies, differing from those of the neutral ground state, for the 2 A 1 (7a 1 1 ) state. In our calculation for the 2 B 1 (2b 1 1 ) state two imaginary frequencies (of b 1 and b 2 symmetry) were obtained. This means that here the symmetric C 2v molecular structure represents a saddle point on the potential energy surface, and one can expect that certain in-plane and out-of-plane distortions of the molecular configuration will lead to structures with lower total energy. Further studies are necessary to ascertain whether this result is an artifact of the method used, and whether the structures with reduced symmetry are bound.

F. Assignment of the spectrum 1. Vertical ionization
The theoretical ionization spectrum of pyridine computed at the ADC(3)/cc-pVDZ level of theory is shown in Figure 4, together with a complete valence shell photoelectron spectrum recorded at a photon energy of 80 eV, and at θ = 0°. The present theoretical spectrum agrees well with previous ADC(3) calculations. 35,47 It reproduces the main features observed in the experimental spectrum and therefore can be used for qualitative assignments. The numerical data for the outer-valence region can be found in Tables III, IV and V. According to the ADC(3) results, the photoelectron band occurring between ~9.1-10.3 eV is formed by the closely spaced transitions into the 1a 2 (π) 1 and 7a 1 (σ N LP ) 1 states.
This is in agreement with the results of the other theoretical methods used in the present work and in the previous studies (Table V). However, the vertical ordering of the 1a 2 (π) 1 and 7a 1 (σ N LP ) 1 states cannot be determined unambiguously since the energy separation is smaller than the accuracy margin afforded by the computational schemes, especially for the 7a 1 (σ N LP ) 1 state, as discussed in Sec. IVC.
Obviously, the best estimates of the vertical ionization energies for the 7a 1 (σ N LP ) 1 and 1a 2 (π) 1 states are provided by the present EOM-IP-CCSDT/cc-pVTZ calculations (9.61 and 9.73 eV, respectively). These values can be improved further by adding the   CBS correction (defined as   = E v (cc-pVZ)  E v (cc-pVTZ)) using the results given in  46 As can be seen from

Adiabatic ionization
We have used the MP2 and CCSD geometries (Tables IX and X) to compute the adiabatic ionization energies and to evaluate the adiabatic energy corrections ( ad ) to the vertical ionization energies  v ( Table V). The  ad corrections are obtained as the differences between the total energies (E tot ) for the cationic state at its optimal geometry (CS) and the ground-state optimal geometry (GS):  ad = E tot (CS)  E tot (GS). The final-state vibrational frequencies were computed at the MP2 level (Table XI), and the corresponding 0-0 corrections ( 00 ) to the adiabatic ionization energies were evaluated (Table V) (Table V).
Our best estimate for the 7a 1 (σ N LP ) 1 0-0 transition energy is 9.31 eV. This energy is still slightly higher than the E 00 values, 9.199, 34 9.197, 36 9.1978, 38 and 9.1215, 39 eV, obtained in recent experiments. It is interesting to note that Riese et al. 38 attempted to characterize the lowest cationic state using density functional theory with a B3LYP functional and the aug-cc-pVTZ basis set. The resulting cationic ground state was predicted to be of 2 B 2 symmetry, with a molecular structure differing considerably from that obtained in our CCSD/cc-pVTZ treatment of the 2 B 2 state.
The best estimate for the 00 transition energy of the 1a 2 (π) 1 state obtained in the present theoretical work is 9.65 eV. The corresponding experimental value is difficult to determine due to the complexity of the lowest photoelectron band, to which this transition contributes. The only experimental value, 9.275 eV 39 , is much lower than our prediction. In view of the very small vertical gap between the 7a 1 (σ N LP ) 1 and 1a 2 (π) 1 states (0.12-0.13 eV at the EOM-IP-CCSDT level of theory), it is very likely that these two states are subject to strong vibronic coupling. This coupling probably also involves the 2b 1 (π) 1 state, which lies 0.7 eV above the 1a 2 (π) 1 state ( Table V) Previous theoretical work has shown that resolved vibrational structure is usually absent above the point of a conical intersection between potential energy surfaces, as here the nuclear dynamics becomes strongly non-adiabatic and give rise to numerous irregular vibronic transitions. 101 (See, for example, Ref. 12 which shows photoelectron spectra of the dichlorobenzenes where vibronic coupling occurs between the lowest two ionic states). In pyridine it seems plausible to suggest that the lack of vibrational peaks above ~10 eV marks the presence of a conical intersection between the 7a 1 (σ N LP ) 1 and 1a 2 (π) 1 states. In such situations, accurate vertical and adiabatic transition energies can be extracted from experimental data only through using appropriate vibronic coupling models based on the results of high-level electronic structure methods. 101,102 The second band in the photoelectron spectrum, with a maximum near 10.5 eV, can be attributed unambiguously to the 2b 1 (π) 1 final state. While the vertical energy of this transition is reproduced rather accurately by most of the theoretical methods (Tables III-V), the absence of any resolved vibrational structure in the experimental spectra indicates a more complex situation that can be understood only within the framework of a suitable vibronic coupling model. Our best estimate for the 2b 1 (π) ionization energy, 10.56 eV, obtained at the EOM-IP-CCSDT level of theory using CBS extrapolation, is in excellent agreement with the measured value (Table III). Our theoretical result for E 00 , 10.30 eV, is also in accord with the 00 transition energy of 10.315 eV reported by Śmiałek et al. 36 The next group of bands is located in the experimental spectrum between ~12 and 15 eV. The observed structure, due to four overlapping bands, is quite complex, with maxima occurring at 12.454, 13.2 13.8 and 14.5 eV. 30 These features correspond to peaks 4-7 in the theoretical spectrum (Figure 4), which, according to our calculations, can be assigned to ionization of the 5b 2 (), 1b 1 (), 6a 1 (), and 5b 2 () orbitals, respectively. The vertical EOM-IP-CCSDT energy for ionization of the 5b 2 () orbital, 12.88 eV (CBS limit), is slightly higher than the measured value (12.454 eV 30 ). A shift in the same direction occurs for E 00 , where our calculated value of 12.31 eV compares with the experimental result of 12.257 eV. 30 The vertical ionization energies, 13.27 and 14.04 eV, of the 1b 1 () and 6a 1 () orbitals, respectively, computed at the ADC(3) level, are in good agreement with the corresponding measurements (Table III).
Our calculations predict that satellites begin contributing to the spectrum above 14 eV.
Several low-intensity 2h-1p transitions, including shake-up satellites associated with the 1a 2 (π), 5b 2 (), and 6a 1 () main lines, appear in the theoretical spectrum ( Figure 4)  respectively, compared to the corresponding main line. According to our ADC(3) results, the intensity of the 1b 2 (π) main line is 0.72, which means that almost 30% of the intensity has been diverted to satellites. While not individually resolved, these satellites increase the width of the observed bands.
The next two bands in the experimental spectrum, with maxima at 15.838 eV and 17.1 eV, 30 are relatively easy to assign. The first is due to ionization of the closely spaced 3b 2 () and 5a 1 () orbitals, and the presence of these two contributions is reflected in the relatively high spectral intensity. 30 The second band can be associated with the 4a 1 () orbital.
Our calculations predict extensive satellite structure in the binding energy range encompassing the 4a 1 () main line, so the observed width will be increased due to these unresolved contributions.
In common with many other organic compounds, the simple one-particle picture of state. This situation is referred to as the "break-down of the orbital picture of ionization". 4

G. Photoelectron angular distributions and photoionization partial crosssections
Photoelectron angular distributions can often be used to help assign the type of orbital (π, σ or non-bonding) associated with a specific band even in congested regions of the spectrum where the individual bands partially overlap. Previous experimental 61,103,104 and theoretical 12,41,105 work has established that anisotropy parameters due to π, σ or non-bonding orbitals exhibit differing energy dependences, and that these differences are particularly evident in the photon energy range ~ 40 -60 eV. The β-parameter corresponding to ionization of a π-orbital tends to lie close to zero at threshold, and then rises rapidly as the photon energy increases to reach a plateau value of at least 1.5 at hν ~ 60 eV. In contrast, the value of an anisotropy parameter associated with a σ-orbital rises much more slowly as the energy increases. In favorable circumstances, these differing energy dependencies allow the type of orbital associated with a specific photoelectron band to be identified through simple comparison of photoelectron spectra recorded either parallel or perpendicular to the polarization vector of the incident plane polarized radiation. Figure 5 shows an example where comparison of perpendicular and parallel photoelectron spectra enable the band due to a π-orbital to be distinguished from neighboring bands associated with σ-orbitals. The band due to the 1b 1 (π) orbital, whose maximum occurs around 13 eV, is much stronger in the spectrum recorded at θ = 0° than it is in that recorded at θ = 90°, because, at a photon energy of 40 eV, the β-parameter of a π-orbital is much higher than that of a σ-orbital. Thus, the band due to the 1b 1 (π) orbital is readily distinguished from those due to the adjacent 5b 2 (σ), 6a 1 (σ) and 4b 2 (σ) orbitals. A full analysis of parallel and perpendicular recordings, extended across all measured bands, allows quantitative experimental anisotropy parameters to be derived as a function of photon energy from our angle resolved spectra.
In previous experimental studies 29,31,34 the variation of the β-parameter, as a function of binding energy, for the outermost photoelectron band of pyridine, encompassing contributions from the 7a 1 (σ N LP ) and 1a 2 (π) orbitals, has been used as an indication of the ordering of these two orbitals. These analyses suggest that the ionization energy of the 7a 1 orbital is lower than that of the 1a 2 orbital. Our present results do not allow such a definite conclusion on the orbital sequence to be reached. Although at some photon energies the value of the βparameter for the low binding energy region of the band is lower than that for higher binding energies, in accord with earlier analyses, 29,31 at other photon energies the opposite variation has been found. Moreover, at many photon energies the β-parameter does not appear to vary significantly across the band. Such a behavior would be consistent with that expected from two substantially overlapping bands whose ionization energies are similar, as our calculations predict, and as previously suggested by Liu et al. 34 It should also be noted that at low photon energies, such as those used in earlier experiments, 29,31 the ionization process may be influenced by autoionization, which is known to affect β-parameter values. Figure 6 shows photoelectron anisotropy parameters for the 1a 2 (π), 7a 1 (σ N LP ), 2b 1 (π) and 5b 2 (σ) orbitals measured in the present experiment together with theoretical predictions.
A complete set of experimental and theoretical β-parameters for all the energy regions specified in Table I is available in the Supplementary Material. In general the agreement between experiment and theory is reasonable and demonstrates the expected differing energy dependencies for ionization from a σ-or π-orbital. The β-parameter for the 2b 1 (π) orbital rises rapidly with increasing energy (as shown by both experiment and theory) in contrast to that for the 5b 2 (σ) orbital where the increase in the β-value is more gradual. Indeed, this contrasting energy variation is evident in the theoretical predictions for the outermost band, comprising contributions from the 7a 1 (σ N LP ) and 1a 2 (π) orbitals. The calculated β-parameter for the 1a 2 orbital increases more rapidly than that associated with the 7a 1 orbital, with the individual curves reaching similar values for a photon energy of ~100 eV. These two contributions are not resolved experimentally and in the photon energy range 20 -50 eV the measured anisotropy parameter lies between those predicted for the 7a 1 and 1a 2 orbitals. It should be noted that the representation of the 7a 1 orbital in these calculations includes none of the post HF refinements discussed above.
The experimentally derived and the calculated absolute photoionization partial crosssections for the 1a 2 (π), 7a 1 (σ N LP ), 2b 1 (π) and 5b 2 (σ) orbitals are plotted in Figure 7. Again, a complete set of results is available in Supplementary Material. The overall agreement between experiment and theory is satisfactory, especially at photon energies above ~30 eV. For some orbitals, the prominent resonances predicted a few eV above threshold appear overestimated.
Such effects have been observed previously in fixed geometry CMS-X calculations. 106 The inclusion of vibrational averaging tends to both reduce the magnitude of the resonance and to increase the peak width.

V. SUMMARY AND CONCLUSIONS
The outer-and inner-valence shell ionization spectra of pyridine were investigated using the ADC(3) approximation scheme for the one-particle Green's function. The vertical ionization energies and the corresponding spectral intensities (pole strengths) were computed, and these were used to interpret the experimental photoelectron spectra. To limit the computational cost, the relatively compact cc-pVDZ basis sets were used in these calculations. These, nevertheless, allow a qualitatively correct description of the spectrum to be obtained.
Outer-valence ionization transitions were also computed using the OVGF and EOM-IP-CCSD methods. Here, the larger aug-cc-pVTZ basis sets were used as these allow the outervalence ionization to be described in a quantitative manner. At the OVGF level, a basis set convergence study was carried out employing the cc-pVnZ and aug-cc-pVmZ (n = D, T, Q, 5, and m = D, T, Q) sets. CBS extrapolation results were obtained.
The theoretical results were found to be in good agreement with the experimental data across the entire binding energy range, and enabled the structure observed in the photoelectron spectra to be assigned. The outer-valence ionic states are well described within the one-electron picture of ionization. The lowest satellite (2h-1p) states appear at about 14 eV, and acquire their intensity from the innermost -orbital. Hence, the spectral intensity of the  1 main state is somewhat decreased, as is typical for heteroaromatic molecules. 61 The ionization of the inner-valence orbitals is influenced by large electron correlation and orbital relaxation effects. These effects are reflected in our calculations in strong 1h and 2h-1p configuration mixing and in the transfer of intensity from the main (1h) state to a manifold of final states. This situation, also referred to as the break-down of the one-electron picture of ionization, 4 occurs in the spectra at binding energies above 20 eV. In the ADC(3) method, the 2h-1p states are treated less accurately (first order) than the 1h states (third order), thereby affording only a qualitative description of the inner-valence spectral regime.

1
), of the pyridine radical cation. The relative positions of the corresponding energy levels, particularly that of the 7a 1 (n) orbital, has been a major issue in the theoretical and experimental literature For a definitive clarification, we have used high-level electronic structure methods, including the EOM-IP-CCSDT scheme, together with the high-quality cc-pVTZ basis set and CBS limit extrapolation. According to our best estimates, the 7a 1 (n) 1 state is the ground state of the pyridine cation, with a vertical ionization energy of 9.72 eV, followed by the 1a 2 () 1 state at 9.85 eV, the 2b 1 () 1 state at 10.56 eV, and the 5b 2 () 1 state at 12.88 eV.
For each of the four lowest cationic states, the equilibrium geometry parameters were computed using the MP2 and CCSD methods, and the harmonic vibrational frequencies were obtained at the MP2 level. These results were used to estimate the adiabatic (0-0) ionization energies (9.31, 9.65, 10.30, 12.31 eV). Except for the 1a 2 () 1 state, the theoretical 0-0 estimates are in accord with the experimental data. For the 1a 2 () 1 state, vibronic coupling to the lower 7a 1 (n) 1 state may play a role, as indicated by the small energy gap between the two states and the rather complex appearance of the 1a 2 () photoelectron band. 34 The orbital relaxation and electron correlation effects in the 7a 1 (n), 1a 2 (), 2b 1 (), and 5b 2 () transitions were analyzed in detail. Often an ionic state is reasonably well described as a 1h HF configuration where the unaffected electrons reside in the ground-state (frozen) HF orbitals. That is the first-order Koopmans description of ionization. At the second-order level, electron correlation and orbital relaxation comes into play, and it is interesting to analyze the corresponding changes in the electron density in the ionic states. To this end, we have computed total electron density difference maps, allowing the frozen and relaxed densities to be compared.
A very strong orbital relaxation effect, amounting to 2.73 eV, was found for the 7a 1 (n) 1 state. This is much larger than the corresponding values of 0. 85 An explicit PT2 expansion of the relaxation energy allowed the relaxation effect in the n lone-pair state to be analyzed in detail. More specifically, by employing the PT2 approach, the main contributions to the relaxation energy in terms of two-electron integrals were identified. The screening of the n positive charge is described by the (|*)  σ π π* J integral reflecting the coupling of the  charge to the density in the highest  and the lowest * orbitals. According to our results, more than 50% of the relaxation energy of the n orbital is due to the -system contribution. This is in agreement with the density difference plots predicting for the n orbital considerable changes in the electron density distribution above the molecular plane. For comparison, the -system contributes only about 20% to the relaxation of 5b 2 () 1 state.
The relaxation mechanism discussed above seems to be rather general and can be present in other heteroaromatic molecules possessing -systems and n-type lone-pairs of the heteroatoms (nitrogen or oxygen). It may be responsible for the near degeneracy between the n and  levels noticed in previous works on such molecules. [49][50][51][52][53][54][55][56][57][58] This will be checked in our future studies.
Photoelectron anisotropy parameters and photoionization partial cross-sections for the outer-valence orbitals have been measured from the ionization threshold to a photon energy of 120 eV. The experimental data have been compared to the corresponding theoretical predictions obtained with the CMS-Xapproach. The overall agreement is satisfactory, thereby supporting the predicted electronic orbital configuration.

SUPPLEMENTARY MATERIAL
See supplementary material for contour plots of pyridine total electron density differences showing the effects of orbital relaxation and electron correlation for ionization from the 7a 1 (σ N LP ), 1a 2 (), 2b 1 (),and 5b 2 () orbitals. Supplementary material also contains a complete set of experimental and theoretical β-parameters, and a complete set of experimental and theoretical absolute photoionization partial cross-sections for all the regions specified in Table I.   Table IV. Vertical ionization energies of pyridine (eV) calculated at the OVGF level of theory using the series of basis sets cc-pVnZ and aug-cc-pVnZ with improving quality (n = D, T, Q, 5), and results for the complete basis set (CBS) limits (n  ). i See text for details (Sec. IV.F.2). j C 2v molecular structure (saddle point, as indicated by the two imaginary frequencies). k See text for details (Sec. IV.F.2). l Obtained using the expression: E 00 = E v (EOM-IP-CCSDT  ) +  ad (CCSD) +  00 (MP2). For the 5b 2 (σ) transition the  00 (MP2) term is omitted, since the MP2 geometry optimization failed for the final state. 54 Table VI. Comparison of Mulliken atomic population in the four highest occupied molecular orbitals of pyridine in its singlet electronic ground state and the pyridine cation radicals in doublet electronic states obtained by ionization of the respective MOs (units are percent of the total orbital populations, equal to two and one, respectively). Calculations performed at the RHF/cc-pVDZ level of theory.

Atom
Population in molecular orbitals Neutral ground state Cation radicals   Figure 1) in its electronic ground-state, 1 A 1 (GS), and the lowest cationic states, 2 A 1 (7a 1

1
), obtained using the CCSD/cc-pVTZ approach (bond lengths in Å, angles in degrees).       The experimental and theoretical (CMS-Xphotoelectron anisotropy parameters of pyridine associated with the 7a 1 (σ N LP ), 1a 2 (π), 2b 1 (π) and 5b 2 (σ) orbitals, corresponding to binding energy regions 1 -3 specified in Table I. The experimental and theoretical (CMS-Xphotoionization partial cross-sections of pyridine associated with the 7a 1 (σ N LP ), 1a 2 (π), 2b 1 (π) and 5b 2 (σ) orbitals, corresponding to binding energy regions 1 -3 specified in Table I. The experimental partial cross-sections have been obtained by combining the measured branching ratios with the absolute photoabsorption cross-section and the photoionization quantum efficiency determined by Tixier et al. 93 See text for details.