| 1. |
- Baev, A., et al.
(författare)
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Doppler interference in dissociative resonant photoemission
- 2002
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Ingår i: Physical Review A. Atomic, Molecular, and Optical Physics. - 1050-2947 .- 1094-1622. ; 66:2
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Tidskriftsartikel (refereegranskat)abstract
- Resonant photoemission involving dissociative core excited states has been the subject of a great number of experimental and theoretical investigations in recent time. The resonant decay of such dissociating systems has been shown to lead to semiatomic Auger electron emission spectra, with particular angular behavior. In the present paper a detailed theoretical analysis of dissociative resonant photoemission spectra of homonuclear diatomic molecules is presented. The theory addresses both fixed in space and randomly oriented homonuclear molecules and emphasizes the Doppler effect and the role of the interference between channels referring to the Doppler split atomic fragments. It is shown that peaks originating from decay in the atomic fragments can be asymmetric and structured due to the Doppler interference effect. The predicted strong non-Lorentzian behavior of the substructure on the top of the Doppler broadened atomiclike contribution is traced to the interplay between decay channels leading to gerade and ungerade final states. Simulations based on wave-packet theory are compared with experimental data for molecular oxygen. Our numerical simulations of the atomiclike resonance of fixed in space molecules show that the spectral profile is very sensitive to the shape of interatomic potentials of core excited and final states. It is shown that the Doppler effect in the decay spectra depends upon the symmetry of the core excited state.
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| 2. |
- Baev, A, et al.
(författare)
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Picturing molecular femtosecond processes through an ultra-fast controllable X-ray shutter
- 2003
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Ingår i: Chemical Physics. - 0301-0104 .- 1873-4421. ; 289, s. 51-56
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Tidskriftsartikel (refereegranskat)abstract
- We show that frequency detuning in a resonant X-ray scattering experiment acts as an X-ray camera shutter by regulating the duration time of the scattering process. The camera shutter can be used to select processes at different time scales for observation. This is illustrated by a resonant Auger study of the ultra-fast dissociation of the core-excited HF molecule. We present experimental results and first principle simulations of the molecular fraction in the resonant Auger spectra of HF which is a dynamical parameter that well illustrates X-ray shutter controlled dissociation.
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| 3. |
- Feifel, R, et al.
(författare)
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Observation of a continuum-continuum interference hole in ultrafast dissociating core-excited molecules
- 2000
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Ingår i: Physical Review Letters. - : AIP Publishing. - 0031-9007 .- 1079-7114. ; 85:15, s. 3133-3136
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Tidskriftsartikel (refereegranskat)abstract
- The femtosecond dissociation of HCl after core excitation has been studied through the resonant Auger decay. The spectra contain contributions from decay occurring at both "molecular" and "atomic" internuclear distances. We have observed a new interference mechanism in these spectra: An atomic spectral line develops into a negative spectral contribution, a "hole," when detuning the excitation energy from the maximum of the Cl 2p(-1)sigma* resonance. Resonant x-ray scattering theory quantitatively explains this behavior as due to a novel destructive continuum-Continuum interference between molecular and atomic contributions to the Auger decay.
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| 4. |
- Gel'mukhanov, Faris, et al.
(författare)
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Vibrationally resolved core-photoelectron spectroscopy as an infinite-slit interferometry
- 2001
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Ingår i: Physical Review A. Atomic, Molecular, and Optical Physics. - 1050-2947 .- 1094-1622. ; 64:1
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Tidskriftsartikel (refereegranskat)abstract
- During a molecular vibration, an atom changes continuously its position. Just as the emitted photoelectron waves, the electromagnetic waves absorbed by the atom in the different positions are strictly coherent and have different well-defined phases. These phases depend on the relation between the instantaneous internuclear distance and the photoelectron, respectively, photon wavelengths. We predict that the interference of these coherent waves strongly influences the vibrational profile of the photoelectron spectra of core electrons in a molecule. This effect increases with increasing x-ray photon frequency and results in a deformation and broadening of the vibrational profile. In the case of surface adsorbed molecules, the vibrational profile depends strongly on the direction of the photoelectron ejection and photon momentum, and the orientational sensitivity of a vibrational profile can even be used as a tool to define the orientation of adsorbed molecules.
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| 5. |
- Gel'mukhanov, Faris, et al.
(författare)
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X-ray Raman scattering under pulsed excitation
- 2000
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 112:13, s. 5593-5603
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Tidskriftsartikel (refereegranskat)abstract
- Accounting for actual and anticipated developments in synchrotron radiation- and in x-ray and free electron laser techniques, we present a theory of resonant scattering of short-pulsed x rays. Both the instantaneous and integrated-over-observation-time x-ray scattering spectra are studied. Contrary to the commonly accepted notion of an instantaneous probability, we find that the correct quantity describing the resonant x-ray scattering (RXS) experiment is the probability integrated over the time of measurement. It is shown that the integral RXS probability is given by the convolution of the RXS probability for monochromatic excitation with the Wiener-Khintchine spectrum of the nonstationary incident radiation. We have found that short pulses yield a new spectral band following a non-Raman dispersion law. The finite duration of pulses results in a broadening of the Raman resonance. Some aspects of the theoretical analysis of the RXS spectra as functions of the pulse duration time are analyzed by numerical calculations.
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| 6. |
- Jansik, Branislav, et al.
(författare)
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Calculations of static and dynamic polarizabilities of excited states by means of density functional theory
- 2004
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 121:16, s. 7595-7600
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Tidskriftsartikel (refereegranskat)abstract
- We present density functional theory and calculations for excited state second order, static or dynamic, properties. The excited state properties are identified from a double residue of a cubic response function. The performance of various functionals, including the generalized gradient approximation and fractional exact Hartree-Fock exchange, is compared to coupled cluster calculations. Applications on excited state polarizabilities of s-tetrazine and pyrimidine show a good agreement with ab initio correlated, coupled cluster, results.
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| 7. |
- Minaev, Boris, et al.
(författare)
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Singlet-triplet transitions in three-atomic molecules studied by time-dependent MCSCF and density functional theory
- 2004
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Ingår i: Molecular Physics. - : Informa UK Limited. - 0026-8976 .- 1362-3028. ; 102:13, s. 1391-1406
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Tidskriftsartikel (refereegranskat)abstract
- Singlet-triplet transition moments and phosphorescence lifetimes have been calculated for the three-atomic molecules HCN, O-3, H2O, H2S, GeF2, GeCl2 and GeBr2 by time-dependent density functional theory (DFT) utilizing quadratic response functions in order to qualify DFT which recently has become available for studies of this kind [TUNELL, I., RINKEVIVIUS, Z., VAHTRAS, O., SALEK, P., HELGAKER, T., and AGREN, H., 2003, J. chem. phys., 119, 11024]. Comparison with ab initio and experimental data indicates that DFT exhibit results of similar quality as explicitly correlated methods which indicates that it indeed is a viable approach for singlet-triplet transitions. O-3 provides an intriguing example in that a systematic investigation of the singlet-triplet transition moment of its Wulf band indicates a clear advantage of the DFT technique despite the multiconfigurational character of the electronic structure of this molecule. The electronic spin-spin coupling and the hyperfine nuclear coupling constants have also been calculated in order to further characterize the triplet state in the spectra of the investigated systems.
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| 8. |
- Piancastelli, M. N., et al.
(författare)
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Bond-distance-dependent decay probability of the N 1s -> pi* core-excited state in N-2
- 2000
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Ingår i: Journal of Physics B. - : IOP Publishing. - 0953-4075 .- 1361-6455. ; 33:9, s. 1819-1826
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Tidskriftsartikel (refereegranskat)abstract
- We report the observation of the unusually weak decay of the N 1s --> pi* core-excited N-2 molecule to the (B) over tilde (2)Sigma(u)(+) final state of N-2(+), which is only detectable in an experiment with high sensitivity. The resonant Auger spectra exhibit an unexpected dependence on the selected vibrational level of the intermediate state. Theoretical calculations show that the interference between direct and resonant photoemission as well as a strong geometry dependence of the decay probability on the bond distance give rise to the observed features.
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| 9. |
- Rinkevicius, Zilvinas, et al.
(författare)
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Restricted density-functional linear response theory calculations of electronic g-tensors
- 2003
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 119:20, s. 10489-10496
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Tidskriftsartikel (refereegranskat)abstract
- A method for calculations of electronic g-tensors based on a spin-restricted open-shell Kohn-Sham formalism and linear response theory is described. Test calculations for main group organic radicals and transition metal compounds have been carried out using two different spin-orbit approximations: Scaled spin-orbit and atomic mean-field spin-orbit operators. The results indicate slightly better performance of the proposed spin-restricted approach compared to previous methods based on the unrestricted Kohn-Sham formalism. An exception to this general improvement are the anions, as they show considerable spin-polarization. The results also show the superiority of the atomic mean field spin-orbit approximation of the spin-orbit operator with respect to the scaled approximation. For main group radicals, quantitative agreement with high level ab initio as well as experimental data are achieved, whereas for transition metal compounds the results systematically underestimate experimental values, showing also a considerable dependency on the employed exchange correlation functional, similarly to previous reports on calculations using unrestricted density functional theory. From this work we conclude that these problems are not primarily associated with the spin-contamination problem, and that they must be referred to the current time-dependent density functional theories as such.
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| 10. |
- Rinkevicius, Zilvinas, et al.
(författare)
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Restricted density functional theory of linear time-dependent properties in open-shell molecules
- 2003
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 119:1, s. 34-46
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we report the derivation and the performance of a spin-restricted density functional formalism for linear time-dependent properties in open-shell molecules. The formalism is based on an exponential parameterization of the density operator with the response functions defined through Ehrenfest's principle. In addition to the derivation of formulas, details of implementation are given as well as a discussion of numerical results for excitation energies and dynamic polarizabilities for a selected set of radicals.
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| 11. |
- Rizzo, A., et al.
(författare)
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Density-functional and electron correlated study of five linear birefringences - Kerr, Cotton-Mouton, Buckingham, Jones, and magnetoelectric - in gaseous benzene
- 2004
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 121:18, s. 8814-8830
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Forskningsöversikt (refereegranskat)abstract
- We present the results of an extended study of five birefringences-Kerr, Cotton-Mouton, Buckingham, Jones, and Magnetoelectric-on benzene in the gas phase. The relevant molecular quantities-first-order properties, linear, quadratic, and cubic response functions-are computed employing the density-functional theory (DFT) response theory, with a choice of functionals. In some cases, different functionals are employed for the wave-function computational step and for the subsequent analytical response calculation to determine the combination yielding at the same time the optimal energy and energy derivative results. Augmented correlation consistent basis sets of double and triple zeta quality are used. The DFT results are compared to those obtained at the Hartree-Fock level and in some cases within a coupled cluster singles and doubles electronic structure model. The study tries to assess the ability of the DFT response theory to describe a wide range of properties in a system of rather large size and high complexity. The relative strength of the five birefringences for plausible experimental conditions is determined and, when possible, comparison is made with the results of the measurements.
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| 12. |
- Rizzo, A., et al.
(författare)
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Sternheimer shieldings and EFG polarizabilities : a density-functional theory study
- 2003
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Ingår i: Chemical Physics Letters. - : Elsevier BV. - 0009-2614 .- 1873-4448. ; 372:04-mar, s. 377-385
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Tidskriftsartikel (refereegranskat)abstract
- The electric field gradient (EFG) at the nucleus, the generalized Sternheimer shielding constants, and the EFG hyperpolarizabilities of a set of reference molecules are computed using analytic density-functional (up to quadratic) response theory. At the three-parameter Becke-Lee-Yang-Parr (B3LYP) level, density functional theory (DFT) underestimates correlation effects compared with other approaches such as coupled-cluster and multiconfigurational self-consistent field. For the prediction of EFG properties of hydrogen nuclei and electron-rich atoms such as halides, DFT/ B3LYP provides results even less reliable than Hartree-Fock theory.
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| 13. |
- Salek, Pawel
(författare)
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A wave-packet technique to simulate resonant X-ray scattering cross sections
- 2003
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Ingår i: Computer Physics Communications. - 0010-4655 .- 1879-2944. ; 150:2, s. 85-98
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Tidskriftsartikel (refereegranskat)abstract
- This article describes algorithms and a program implementation for wave packet calculations of resonant X-ray scattering cross sections of molecules with one active internal degree of freedom. The program uses a time-dependent formalism and a grid representation of the wave packets. The potentials of ground, core-excited and final states can be specified by analytical expressions or by discrete sets of energies on arbitrary grids allowing for interfacing with electronic structure packages. The theory on which the program is founded is briefly reviewed. The implemented numerical algorithm is described in detail together with usage instructions and a sample application.
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| 14. |
- Salek, Pawel, et al.
(författare)
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Calculations of two-photon absorption cross sections by means of density-functional theory
- 2003
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Ingår i: Chemical Physics Letters. - 0009-2614 .- 1873-4448. ; 374:06-maj, s. 446-452
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Tidskriftsartikel (refereegranskat)abstract
- We present density-functional theory and calculations for two-photon absorption spectra of molecules. The two-photon absorption cross sections are defined in terms of the single residues of the quadratic response function, which was recently derived for density-functional theory using the time-dependent variation principle and the quasi-energy ansatz. The cross-section dependence on different functionals, including the general gradient approximation and hybrid theory, is examined for a set of small molecules. The results of hybrid density-functional theory compare favorably with those from singles-and-doubles coupled-cluster response calculations.
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| 15. |
- Salek, Pawel, et al.
(författare)
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Density-functional theory of linear and nonlinear time-dependent molecular properties
- 2002
-
Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 117:21, s. 9630-9645
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Tidskriftsartikel (refereegranskat)abstract
- We present density-functional theory for linear and nonlinear response functions using an explicit exponential parametrization of the density operator. The response functions are derived using two alternative variation principles, namely, the Ehrenfest principle and the quasienergy principle, giving different but numerically equivalent formulas. We present, for the first time, calculations of dynamical hyperpolarizabilities for hybrid functionals including exchange-correlation functionals at the general gradient-approximation level and fractional exact Hartree-Fock exchange. Sample calculations are presented of the first hyperpolarizability of the para-nitroaniline molecule and of a porphyrin derived push-pull molecule, showing good agreement with available experimental data.
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| 16. |
- Salek, Pawel, et al.
(författare)
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Doppler effect for bound nuclear motion and its manifestation in resonant photoemission of oriented systems
- 2000
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Ingår i: Chemical Physics Letters. - 0009-2614 .- 1873-4448. ; 328:4-6, s. 425-430
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Tidskriftsartikel (refereegranskat)abstract
- In contrast to the common notion that Doppler effects have relevance only for atoms or molecules in free motion, we show that such effects should clearly be observable - and accounted for - in resonant scattering also when the nuclei are in bound state quantized motion. This untrivial effect does not influence the position of single resonances but only the center of gravity of the spectra. It is found to strongly depend on the duration time of the scattering.
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| 17. |
- Salek, Pawel, et al.
(författare)
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Dynamical properties of X-ray Raman scattering
- 2003
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Ingår i: Physical Chemistry, Chemical Physics - PCCP. - : Royal Society of Chemistry (RSC). - 1463-9076 .- 1463-9084. ; 5:1, s. 1-11
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Tidskriftsartikel (refereegranskat)abstract
- This paper reviews theory for resonant X-ray scattering, emphasizing the dynamical aspects related to the nuclear motion during the scattering process. Various features of the theory are illustrated by simulations on small molecules in the gaseous phase. After the introduction de ning the X-ray Raman scattering process, we briefly describe some central concepts of its theory: the Raman law, the time hierarchy of X-ray scattering and the generalized Kramers-Heisenberg relation. We then go into some detail of various theoretical aspects of the process: transition moments and light polarization, time-dependent formulation and wave packets. We review two aspects of recent experimental focus: Doppler effects and generalized Franck-Condon factors. In the last part of the paper we discuss X-ray Raman scattering of dissociative core excited states, and the intriguing concept of a duration time for the scattering. We address dissociative resonant photoemission from first principles, with particular emphasis on the conditions for observing so-called atomic peaks and atomic holes. The atomic holes are the results of continuum continuum interference effects between the atomic and molecular channels which may act destructively under certain conditions. We demonstrate that the resonant contribution and the evolution of the atomic peaks can be subject to strong dynamical suppression caused by nuclear motion. In general, the molecular geometry dependence of the electronic transition moments can change the scattering cross section quite dramatically. This is exemplified here by an unusual spectral flattening for the B state of N-2, and by a breakdown of the spectator versus participator classification in a part of the non-radiative resonant photoemission spectrum of HF. The notion of a duration time in resonant X-ray scattering is put forward as a powerful concept for predicting and understanding qualitative aspects of many of the effects or processes treated in this paper. The role of the pulse shape for short pulse X-ray excitation is also discussed.
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| 18. |
- Salek, Pawel, et al.
(författare)
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Dynamical suppression of atomic peaks in resonant dissociative photoemission
- 2001
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Ingår i: Chemical Physics Letters. - 0009-2614 .- 1873-4448. ; 343:3-4, s. 332-338
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Tidskriftsartikel (refereegranskat)abstract
- Resonant excitation to the F1s-sigma* dissociative state in hydrogen fluoride gives a photoelectron spectrum where the spectator part contains strong atomic lines but a participator part where such lines are lacking. We demonstrate that this contrasting behaviour between the two parts is due to a strong dynamical suppression of the resonant contribution, making direct main state photoionization the dominating channel even at resonant conditions.
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| 19. |
- Salek, Pawel, et al.
(författare)
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Resonant x-ray Raman scattering involving avoided crossings in the final-state potential-energy curves
- 2000
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Ingår i: Physical Review A. Atomic, Molecular, and Optical Physics. - : Elsevier BV. - 1050-2947 .- 1094-1622. ; 62:6
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Tidskriftsartikel (refereegranskat)abstract
- The role of avoided crossings for resonant x-ray Raman scattering is explored. The strong dependence of the electronic transition amplitudes on internuclear distance at avoided crossings causes new, and even anomalous, spectral features, and can bring about a breakdown of the generally accepted participator or spectator classification of the transitions. We report, and apply the theory on, an experimental observation of the electron decay of the N 1s-->pi* core-excited level in N-2 the B(2)Sigma (+)(u) final state with peculiar, non-Franck-Condon, vibrational intensity distributions.
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| 20. |
- Salek, Pawel, et al.
(författare)
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Theory and first principles calculations of dissociative resonant photoionization : The evolution of atomic peaks and holes
- 2002
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 116:2, s. 629-645
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Tidskriftsartikel (refereegranskat)abstract
- We present theory and computational method for analyzing dissociative resonant photoemission from first principles. Particular emphasis is devoted to the conditions for observing so-called atomic peaks and atomic holes. The atomic peaks are connected with photoemission following resonant excitation to dissociative core excited states which show signals from scattering channels involving the dissociation (atomic) fragments in addition to those involving the compound molecule. The holes are the results of continuum-continuum interference effects between these two, atomic and molecular, channels which may act destructively under certain conditions. We apply a novel electronic structure method to compute the transition moments for the resonant and direct photoemission channels including their dependence on internuclear distances and their interference. The relevant matrix elements involving the photoelectron are obtained using similar techniques for the two types of channels, with the scattered electron wave in each case being determined in the full molecular anisotropic potential. A study of resonant photoemission through the core excited sigma* states of HF and HCl indicates that the appearance of the atomic peaks and holes is subtly dependent on the nuclear dynamics, the potential energy curves, and the excitation photon frequency. We demonstrate that the resonant contribution and the evolution of the atomic peaks can be subject to strong dynamical suppression, so strong in fact that main state photoionization may constitute the dominating channel even at resonant conditions. It is shown that such dynamical suppression explains that resonant excitation to the F 1s-sigma* dissociative state in hydrogen fluoride gives a photoelectron spectrum in which the spectator part contains strong atomic lines but a participator part where such lines are lacking, although they both refer to the same, dissociative, core excited state. The findings in the present work give evidence that both direct and resonant channels should be simultaneously considered in analyses of the dissociative photoemission process even at resonant conditions.
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| 21. |
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| 22. |
- Thogersen, L., et al.
(författare)
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The trust-region self-consistent field method : Towards a black-box optimization in Hartree-Fock and Kohn-Sham theories
- 2004
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 121:1, s. 16-27
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Tidskriftsartikel (refereegranskat)abstract
- The trust-region self-consistent field (TRSCF) method is presented for optimizing the total energy E-SCF of Hartree-Fock theory and Kohn-Sham density-functional theory. In the TRSCF method, both the Fock/Kohn-Sham matrix diagonalization step to obtain a new density matrix and the step to determine the optimal density matrix in the subspace of the density matrices of the preceding diagonalization steps have been improved. The improvements follow from the recognition that local models to E-SCF may be introduced by carrying out a Taylor expansion of the energy about the current density matrix. At the point of expansion, the local models have the same gradient as E-SCF but only an approximate Hessian. The local models are therefore valid only in a restricted region-the trust region-and steps can only be taken with confidence within this region. By restricting the steps of the TRSCF model to be inside the trust region, a monotonic and significant reduction of the total energy is ensured in each iteration of the TRSCF method. Examples are given where the TRSCF method converges monotonically and smoothly, but where the standard DIIS method diverges.
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| 23. |
- Tunell, I, et al.
(författare)
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Density functional theory of nonlinear triplet response properties with applications to phosphorescence
- 2003
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 119:21, s. 11024-11034
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Tidskriftsartikel (refereegranskat)abstract
- We present density functional response theory generalized to triplet excitations. A method based on an exponential parametrization of the spin-dependent density operator is derived for the evaluation of linear and quadratic response functions for spin-dependent perturbations. The developed methodology is applicable to commonly available functionals, also hybrid functionals including exchange-correlation functionals at the general gradient-approximation level and fractional exact Hartree-Fock exchange. Illustrative calculations are presented for singlet-triplet transition moments and phosphorescence lifetimes, providing numerical data on these quantities for the first time using time-dependent density functional theory.
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| 24. |
- Watson, M. A., et al.
(författare)
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Linear-scaling formation of Kohn-Sham Hamiltonian : Application to the calculation of excitation energies and polarizabilities of large molecular systems
- 2004
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 121:7, s. 2915-2931
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Tidskriftsartikel (refereegranskat)abstract
- We present calculations of excitation energies and polarizabilities in large molecular systems at the local-density and generalized-gradient approximation levels of density-functional theory (DFT). Our results are obtained using a linear-scaling DFT implementation in the program system DALTON for the formation of the Kohn-Sham Hamiltonian. For the Coulomb contribution, we introduce a modification of the fast multipole method to calculations over Gaussian charge distributions. It affords a simpler implementation than the original continuous fast multipole method by partitioning the electrostatic Coulomb interactions into classical and nonclassical terms which are explicitly evaluated by linear-scaling multipole techniques and a modified two-electron integral code, respectively. As an illustration of the code, we have studied the singlet and triplet excitation energies as well as the static and dynamic polarizabilities of polyethylenes, polyenes, polyynes, and graphite sheets with an emphasis on the trends observed with system size.
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| 25. |
- Watson, M. A., et al.
(författare)
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The calculation of indirect nuclear spin-spin coupling constants in large molecules
- 2004
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Ingår i: Chemistry - A European Journal. - : Wiley. - 0947-6539 .- 1521-3765. ; 10:18, s. 4627-4639
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Tidskriftsartikel (refereegranskat)abstract
- We present calculations of indirect nuclear spin-spin coupling constants in large molecular systems, performed using density functional theory. Such calculations, which have become possible because of the use of linear-scaling techniques in the evaluation of the Coulomb and exchange-correlation contributions to the electronic energy, allow us to study indirect spin-spin couplings in molecules of biological interest, without having to construct artificial model systems. In addition to presenting a statistical analysis of the large number of short-range coupling constants in large molecular systems, we analyse the asymptotic dependence of the indirect nuclear spin-spin coupling constants on the internuclear separation. In particular, we demonstrate that, in a sufficiently large one-electron basis set, the indirect spin-spin coupling constants become proportional to the inverse cube of the internuclear separation, even though the diamagnetic and paramagnetic spin-orbit contributions to the spin-spin coupling constants separately decay as the inverse square of this separation. By contrast, the triplet Fermi contact and spin-dipole contributions to the indirect spin-spin coupling constants decay exponentially and as the inverse cube of the internuclear separation, respectively. Thus, whereas short-range indirect spin-spin coupling constants are usually dominated by the Fermi contact contribution, long-range coupling constants are always dominated by the negative diamagnetic spin-orbit contribution and by the positive paramagnetic spin-orbit contribution, with small spin-dipole and negligible Fermi contact contributions.
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