CHIROPTICAL PROPERTIES
OF PLATINUM(II) COMPLEXES:
A COMPUTATIONAL STUDY
BY
EMMANUELE CANNAVÒ
SUPERVISOR:
CONTENTS
Contents p. 3
Summary p 5
Introduction p. 7
State of the art p. 11
Theoretical methods p. 25
The Gaussian suite p. 45
Simulations p. 53
Conclusions p. 97
SUMMARY
The aim of the present work is the computational analysis, by means of electronic spectroscopy, of the organometallic Platinum (II) complex [PtLCN
2] (where L
is referred to the achiral heterobidentate chelate ligand 2-((2-((2-phenyl) ethynyl) phenyl) ethynyl) pyridine, which bounds to the metal centre via a carbon (C) and nitrogen (N) atom); this chemical species has previously been studied, both from the theoretical and experimental points of view, by Schulte and co-workers in 2017 (for more details on that article, see ref. 45 and the relative supplementary information paper).
The computational protocol adopted for this purpose is based on the Density Functional Theory (DFT) formalism and its time-dependent extension, TD-DFT, for the treatment of the time-dependent many-body problem, together with Polarizable Continuum Model (PCM) in its integral equation formalism (IEF-PCM), in order to account for the solvation effects. The most prominent feature of the protocol (and its main innovation with respect to the abovementioned previous work) is the computation of vibrationally resolved (or vibronic) electronic spectra, that allow for a better and deeper understanding of the optical characteristics of the substance object of this study. The two spectroscopies examined were One-Photon Absorption (OPA) and Electronic Circular Dichroism (ECD).
The protocol, if successful in its spectroscopic predictions, will hopefully be suitable for the simulation of chiroptical properties of other similar organometallic compounds (obtained, for example, by changing the number and type of the ligands bounded to the Pt centre, and possibly still to be synthesized) and could potentially be extended to include other kinds of spectroscopies, such as One-Photon Emission (OPE), Resonance Raman (RR) and Two-Photon Absorption (TPA).
The thesis opens with a short introduction, where first the central role of organometallic compounds in the realization of cutting-edge photovoltaic devices is outlined; the subject of chirality and its importance in applications is then briefly treated. Subsequently, a primer in vibronic spectroscopy, the workhorse of the protocol, is given; finally, an explanation of why computational chemistry can provide invaluable tools in the analysis and characterization of chemical compounds is offered. The second chapter is devoted to the state of the art in the rich research field of organometallic compounds. The first paragraph covers their applications to nonlinear optics; the second paragraph is about Dye-Sensitized Solar Cells (DSSCs), in which organometallic complexes are mainly involved as dyes; the attention is then
shifted to the Platinum complexes: an ample third paragraph is dedicated to the compounds this metal forms with bidentate, tridentate and other types of ligands; finally, a longer exposition on the current trends in vibronic spectroscopy is given.
The third chapter describes the theoretical methods that have been used in this work, starting with an exposition of (TD)DFT, explaining its strengths and advantages. A concise history of DFT is then presented, from the Hohenberg-Kohn theorem and Kohn-Sham equations to double hybrid functionals; an exposition on PCM for the treatment of solute-solvent interactions and its formulation in terms of integral equations follows and the chapter concludes with a paragraph on the theory of vibronic spectroscopy and its different concepts.
The fourth chapter is the computational section of the thesis: the Gaussian suite, used to implement the various calculations, is presented, and the meaning of a series of its keywords thoroughly explained.
The simulations chapter contains the results of this computational study. Approximately in this order, it features: a description of the preliminary steps to be done before the vibronic calculation (namely, the geometry optimizations and frequency calculations for both the ground and the excited state, and the force computation for the vertical gradient spectra); a comparison of the purely electronic computed spectra with the digitalized experimental one, done both for OPA and ECD spectra with all the functional/basis set pairs; a band matching process for assigning the transitions to the respective experimental bands; a comparison with adiabatic hessian (both Franck-Condon and Franck-Condon-Herzberg-Teller), vertical gradient and adiabatic shift (Franck-Condon only) spectra, for a specific functional/basis set pair; a second benchmark against the experimental data, but this time using only vertical gradient vibronic spectra, again calculated for all functionals and basis sets at the Franck-Condon level; another plot with vertical gradient calculations on different excited states juxtaposed with the experimental data.
The chapter presents also a detailed (where possible) description and explanation of problems encountered during this work (some of which are common to this kind of systems) and possible ways of solving them.
INTRODUCTION
PHOTOVOLTAIC DEVICES
Organic Photovoltaic (OPV) devices are now entering a new era in which the limits of silicon structures become crucial for increasing the external quantum efficiency of the solar cells. This classical c-Si bulk material is mainly used as crystalline structure and can be prepared as ingots, ribbons, and wafers. In this category of crystalline silicon solar cells, a special class of materials was developed also known as p-n junction. The c-Si based solar cells exhibit a limited efficiency at around 20%.
More recently, the newest OPVs are based on donor-acceptor heterojunctions employing new organic materials like perovskites, for which the efficiency was improved from 3% to more than 20% starting with 2009. These new solar cells, based on perovskites as active layers, have the main disadvantages of the long-term instability (in practice, they undergo a quick degradation after a few working days). The replacement of the organic-inorganic structure, which includes the perovskite crystals, with more stable chemical compounds becomes a significant issue in this field. The stability problem could be overcome by using alternatives like the organometallic compounds which are very interesting from the scientific and technological points of view due to their long stability as they were already used in the OLED’s technologies for a long time. The advantages of organometallic compounds are connected with the presence of the ionic metals in molecules which induces some interesting properties such as:
a) architectural templates- extended molecular architectures could be developed which induces better electrical conductivities;
b) active redox centres- the possibility of modifying the electronic properties that influence the electrical conductivity of the molecules;
c) tuning of the HOMO-LUMO bandgap- the molecular orbitals from the
metal-ligand hybridization together with the addition of functional groups can tune the bandgap of these compounds.
CHIRALITY
Chirality can be found almost everywhere in the natural world: chiral molecules, such as proteins, nucleic acids and ribonucleic acids, constitute the basic elements of all living systems, which are almost exclusively composed of elementary entities with single optical activity, and play a very important role in various physiological processes. At the same time, these chiral molecules have found widespread use in many different areas, such as chiral recognition, enantioseparation and asymmetric synthesis (a particular type of chemical synthesis, mainly employed in the pharmaceutical industry, in which new elements of chirality are formed in a substrate molecule). Thanks to their characteristic features of optical rotation, selective emission of polarized light, and circular dichroism, circularly polarized luminescent materials have recently attracted extensive attentions, exhibiting a wide range of optoelectronic applications, such as optical data storage (CD, DVD and Blue-Ray technologies), liquid crystal display panels, 3D imaging systems, nonlinear optics, spintronic (spin electronic) devices and biological probes.
According to the difference of light wave vibration orientation vector in the direction perpendicular to the propagation of light wave, polarized light can be classified into different kinds, including linearly, circularly, elliptically, and partially polarized light. Among them, circularly polarized light improves the quality of the image contrast and protects sight health. Generally, it could be produced from plane polarized light via the use of a quarter wave plate, while it causes a loss in energy during the transition process from nonpolarized light to plane polarized light.
Therefore, there is a clear interest to develop a series of chiral nonracemic luminescent materials enabling to directly generate circularly polarized luminescence (CPL) containing right- and left-circularly polarized light. In CPL spectra, the differential emission intensity of right- and left-CPL is quantified by the luminescence dissymmetry factor gL, that can potentially be a function of many different parameters, such as type of solvent, temperature, concentration and orientation of the substance in the solution, presence of free ions. To date, many different types of chiral luminescent systems, including small organic luminophores, conjugated polymers, supramolecules, liquid crystals and organometallic complexes, have been developed; in particular, this last class of composites shows the useful feature of being able to modify its optical properties by means of changing the type and/or number of ligands.
VIBRONIC SPECTROSCOPY
Optical spectroscopy is one of the main methodologies for the identification and characterization of molecular excited states: alongside more standard spectroscopic techniques, such as one-photon absorption (OPA) and one-photon emission (OPE), more recent developments, like resonance Raman (RR) and two-photon absorption (TPA), are increasingly employed to describe in an extensive way the chemical properties of molecules: RR is particularly interesting for large or complex molecules, since it is possible to choose a transition localized on the moiety of interest and obtain a selective enhancement (by up to a factor of 105-106) of the vibrations connected to
this region. Due to the two successive transitions, TPA can provide hindsight on a local excited electronic structure.
Additionally, because of the progress in the development of chiral electronic spectroscopies, for example electronic circular dichroism (ECD) and circularly polarized luminescence (CPL), chiral systems too can be accurately studied. The analysis and interpretation of the wealth of information available from spectral data can be eased using quantum mechanical computational tools, DFT and its time-dependent extension in particular.
It is also of foremost importance to be able to develop high quality theoretical methods to simulate the shape of the electronic spectrum without an excessive reliance on phenomenological parameters; in this way, it is possible to obtain predictions that can be readily compared with experimental data, thus making the extraction of valuable information from them a faster and easier feat. The shape of the spectrum is influenced by the dynamic behaviour of the nuclei of both the system under study and the surrounding environment (often, solvent molecules); in order to get the best results in terms of accuracy, it is necessary to account for the quantum-mechanical nature of the molecular vibrations. The appearance of vibrational progressions (a series of transition with a common energy level) is determined by quantum effects, which also play an important role in determining the width of the transition bands. In addition, while the Franck-Condon principle has a well-established interpretation in classical physics, in order to move beyond it (by means, for example, of the Herzberg-Teller approximation) to better predict the line intensities, a quantum-mechanical vibronic approach is needed.
Computationally speaking, the theoretical methods developed for simulating vibrationally resolved electronic spectra are of two main types: in the time-independent (TI) formulations, the spectrum is made up by a sum of individual vibronic transitions that are analysed independently one from the other; on the other hand,
time-dependent (TD) methods compute the final spectrum as the Fourier transform of the transition dipole moment autocorrelation function.
WHY COMPUTATIONAL CHEMISTRY?
As already mentioned above, one of the main strengths of organometallic compounds lies in the possibility to accurately tune the optical properties thanks to the variety of ligands available; however, in order to be able to operate in an informed manner, it is necessary to have an adequate knowledge of the chemical and physical properties of the compounds themselves. To this end, various types of spectroscopy (emission, absorption, scattering among others) can provide a considerable amount of information, even to the point of making the interpretation of experimental data problematic. This is where the models provided by computational chemistry come into play: these theoretical structures can be, in certain cases, lead to results that can be almost as accurate as to rival the results of the experiments. However, their computational weight increases considerably with the size of the system, so it is necessary to find a compromise between accuracy and speed of calculations.
STATE OF THE ART
NONLINEAR OPTICS
Organometallic compounds have shown remarkable potential as a third order nonlinear material, thanks both to the charge transfer phenomenon between the ligands and the metal and the switchable nonlinearity, the last linked to multiple electronic states of the central metal atom. It has been demonstrated that the two abovementioned properties allow for significant values of first molecular hyperpolarizability.
In ref. 44, the authors present the third harmonic generation response of Znq2
(Bis- (8-hydroxyquinolinato)zinc), Cuq2 (8-Hydroxyquinoline copper(II)), and Alq3
(Tris-(8-hydroxyquinoline)aluminium) organometallic compounds (figure. 1).
These kinds of compounds are different from others as, apart from π-π transitions that can be found in the organic system, they can also exhibit intramolecular charge transfer (ICT) from ligand to metal (LMCT), from metal to ligand (MLCT), or d-d transitions. In coord-dination complexes d-d orbitals of transition metals can interact with the conjugated π-electron orbitals of the ligand. This phenomenon clearly enhances the possibility of a third order susceptibility tailoring process and better design of molecules for nonlinear optics purposes.
Figure 1 Chemical structures of investigated molecules: (a) 8-Hydroxyquinoline zinc [Znq2], (b)
An experiment was conducted for s and p polarizations of incident beam (that is, with the electric field respectively perpendicular and parallel to the incidence plane), using the Maker fringes technique. The third order nonlinear susceptibility was estimated using the Kubodera and Kobayashi comparative model, based on the fact that the compounds exhibit high linear absorption of the generated third harmonic wavelength (355 nm). These complexes were deposited as thin films by means of physical vapor deposition (PVD). The studied complexes vary in terms of the coordination centre and number of quinoline ligands, which visibly influence their nonlinear response. The global hybrid B3LYP functional with the basis set 6-31G(d) was used in computing the linear and non-linear optical properties. The computed γtot
value (second order total hyperpolarizability) is superior to that of methylene blue and the theoretical results agree quite well with the experimental ones.
DYE-SENSITIZED SOLAR CELLS
The authors of ref. 40 have performed calculations on the panchromatic N749 dyes adsorbed on the (TiO2)28 surface. N749 is a prototypical form of Ru(II) complexes
for dye sensitized solar cells (DSSCs), which possesses a terpyridine tridentate ligand with four different protonation states (0, 1, 2, or 3 carboxylic protons). Depending on the type of proton bonding interaction (protonated or deprotonated), seven N749/(TiO2)28 surface models have been applied in the study for the geometry
optimization, frontier molecular orbital level diagrams and calculated absorption spectra. The moderate surface area of the (TiO2)28 cluster is optimal for N749 dyes
adsorbing behaviours so that all calculations can still be performed at a quantum mechanical level.
The calculated absorption spectra of these seven various N749/(TiO2)28 models
(see figure 2) are in good agreement with the experimental results, with onset ranging from the visible to near-IR region. The combination of the adsorption energy onto TiO2
and calculated absorption spectra shows that the deprotonated dyes constitute the most favourable and dominant structure in the DSSC devices. The frontier molecular orbital graphs (figure 3) indicate that the electron charge distributions of all HOMOs are located at the N749 dyes, while LUMOs are localized at the (TiO2)28 surface or
delocalized at the interfacial regions of N749/(TiO2)28. The corresponding transitions
are thus more similar to a type of optical electron transfer, injecting the electron to the interfacial TiO2.
Figure 2 The experimental N749-1H/TiO2 and calculated absorption spectra of the three
N749-P/(TiO2)28 surface models in acetonitrile. The calculated spectra are obtained by a Gaussian convolution
with σ = 0.19 eV. Their intensities have been rescaled to match their absorption maxima over 560 nm. The violet, orange, and dark yellow vertical lines at the bottom of the graph represent the relative oscillator strengths. (from ref. 40).
The accurate ab initio modelling of prototypical and well-representative photoactive interfaces for candidate dye-sensitized solar cells (DSSCs) is a perennial problem in physical chemistry. To this end, the authors of ref. 13 studied the reliability of a number of exchange-correlation functionals in DFT calculations, (which then paved the way to a more extensive dynamic analysis by means of ab initio molecular dynamics (AIMD)), in predicting the behaviour of a system mimicking the essential workings of a DSSC: the energetic properties of a [bmim]+[NTf2]−room-temperature ionic liquid (RTIL) solvating an N719-sensitizing dye adsorbed onto an anatase−titania (101) surface were scrutinized. Operating this way, important insights into how using an RTIL as electrolytic hole acceptor alters and modulates the dynamical properties of the widely used N719 dye have been obtained. A fully crossed study has been carried out comparing the Becke−Lee−Yang−Parr (BLYP) and Perdew−Burke− Ernzerhof (PBE) functionals, both unsolvated and solvated by the RTIL, both with and without Grimme D3 empirical dispersion correction terms (see figure 4). Furthermore,
Figure 3 Left: frontier molecular orbitals of HOMO and LUMO+6 in N749-1H-DP/(TiO2)28 during optical
absorption photoexcitation. Right: charge density difference between the ground state S0 and the
excited state S2. Pink mesh indicates the decrease of charge density, while green mesh indicates the
vibrational spectra for the photoactive interface in the DSC configuration were calculated by
Figure 4 Frontal view showing the relaxed geometries of the BLYP and PBE systems. Carbon is shown in dark grey, nitrogen in dark blue, oxygen in red, hydrogen in white, titanium in light grey, sulphur in yellow, and ruthenium in light green. System types: I, unsolvated; II, solvated with [bmim][NTf2]; and III, solvated with [bmim][NTf2] with dispersion corrections. Solvated systems are shown without solvent, for ease of viewing (from ref. 13).
means of Fourier-transforming atomic mass-weighted velocity autocorrelation functions.
The ab initio vibrational spectra were compared to high-quality experimental data and against each other; the effects of various methodological choices on the vibrational spectra were also studied, with PBE generally performing best in producing spectra, which matched with the expected experimental frequency modes.
PLATINUM COMPLEXES
In the last few years, the combination of transition metal atoms with organic molecules has been widely employed as a very interesting choice in developing of the CPL-active materials for applications in the field of photonic. The diversity of the coordination modes, ranging from the octahedral configuration of d6 metal complexes,
the square configuration of d8 metal complexes, to the triangular configuration and
linear geometries of d10 metal complexes led to a wider variety of structures for these Figure 5 Chemical structures of Pt(II) complexes (from ref. 29)
transition metal complexes. Furthermore, the excited states have provided the complexes with interesting luminescence properties. Moreover the molecules can easily be combined with the help of metal–metal interactions and/or various noncovalent interactions, including hydrogen bonding, π–π stacking, and hydrophobic–hydrophobic interactions.
In the following paragraphs are described the materials based on Pt(II) complexes which were found to present a square-planar geometry with four coordinating structures. Simultaneously, by changing the chelated number of ligands or modulating structure of complexes, the emission colour could be tuned from the visible region to near-infrared, owing to the Pt–Pt interaction, which makes them excellent optoelectronic materials.
Complexes with bidentate ligands
[n]Helicenes are polycyclic π-conjugated molecules formed of n ortho-fused aromatic rings with intrinsic helical chirality. This unique screw-shaped π-conjugated structure endows [n]helicenes with outstanding optical properties, such as large optical rotation, strong CPL signals and intense emission. Metal-based helicenes have emerged as promising candidates for photoelectronic applications, such as OLEDs, switches, sensors, etc., due to the various advantages of the spin–orbit coupling of heavy-metal ions and electronic interactions between metal centre and ligand. So far, the incorporation of Pt(II) ion into helical backbone is the main method for the establishment of CPL-active Pt(II) systems.
Helicene derivatives with Pt(II) incorporated into ortho-annulated π-conjugated backbones were first prepared in 2010. Pt(II) helicenes with a square-planar geometry were synthesized by means ortho-metalation and exhibited red phosphorescence with luminescence efficiency up to 10%. The mirror-image CD spectra of metallahelicenes indicated that the chirality of complexes was inherited from helicene derivatives. When using enantiopure P/M Pt(II) helicenes to react with iodine in dichloromethane (CH2Cl2), the corresponding Pt(IV) helicenes (46–48) were obtained with octahedral
geometry.
The intensity of CD signals (45-P) exhibited a decrease at high energy and an inversion at 280 nm upon the treatment with iodine and could return to the original state (45-P) after adding zinc powder. With this strategy, a series of enantiopure mono-cycloplatinated-[n] helicene derivatives (49–52) that bear 2-phenylpyridine ligands have been designed, which showed efficient red emission with φ over 7%, and relatively low |glum| values about 0.005 (49 and 50) compared with that of compound.
Based on the above-mentioned results, the idea of utilizing Pt(II) helicenes as phosphorescent dopants in a single layer solution-processed CP-OLEDs was
adopted. The CP-OLEDs using enantiopure 45-M doped into OXD-7:PVK (7:3, w/w) exhibited a turn-on voltage of 3.6 V, the maximum luminance of 374 cd m−2 and a maximum power and current efficiency of 0.23 lm W−1 and 0.49 cd A−1, respectively, which were comparable to those of the devices using racemic 45 (5.7 V, 230 cd m−2, 0.16 lm W−1, and 0.52 cd A−1). Importantly, the CP emission of these two enantiomers exhibited opposite signals and the gEL factors of 45-P and 45-M at the emission
maximum were +0.22 and −0.38, respectively (Figure 5c). The high |gEL| values of
devices are more than 30-fold larger than the values recorded for 45-M/P in solution. In other studies, upon the treatment of protonation, helicene–bipyridine cycloplatinated derivative was obtained from the nonprotonated 53, along with the increasing intensity of emission and the amplification of CPL signals at 547 nm, which was ascribed to the delocalization of the extended π-conjugated systems.
Additionally, diastereoisomeric bis-cycloplatinated complexes (54–57) with two possible arrangements of heterochiral (P,M) and homochiral (P,P)/(M,M) are reported to efficiently enhance the chiroptical properties of helicene derivatives, due to the strong interaction between the metal-bridged π-helices.
Complexes with tridentate ligands
The substitution of bidentate ligands with helical tridentate ligands provides a novel strategy to realize the chirality of Pt(II) complexes. Square planar cyclometalated Pt(II) complexes have been proved to be excellent candidates as substitutions of Pt(II) helicenes. Compared with the cyclometalated Pt(II) complexes bearing bidentate C^N ligands, the tridentate C^N-cyclometalated Pt(II) complexes possess a more rigid square planar geometry, resulting in the decrease of nonradiative transition through restricting the distortion of complex.
Intense inter- or intra-molecular Pt–Pt interactions were found in some square planar cyclometalated Pt(II) complexes. The interaction between the dz2 orbitals of the
complexes leads to a new d(σ*–π*) transition. Such a transition may possess lower energy than that of the π–π* transition. For example, it was demonstrated the design of different tridentate cyclometalated Pt(II) complexes (58–61 in Scheme 3) using C^N^N and N^C^N type ligands with chiral (R/S) pinene as main ligands. A shorter Pt–Pt distance (2.886 Å) was found in trinuclear complexes 60-P and 60-M, indicating that strong Pt–Pt interactions existed in these complexes, while Pt–Pt interactions were absent in other mononuclear complexes (58-P/M and 59-P/M). Interestingly, complexes 58-P/M with slightly distorted square-planar geometry are dissociated in methanol solutions and the chirality was rather weak closing to zero. The complexes 58-P/M showed aggregation behaviours in aqueous solutions, forming 1D helical chain structures through Pt–Pt, π–π, and hydrophobic- hydrophobic interactions, while
enhancing the chiral environment with the glum values of −0.0018/+0.0012 around the
maximum emission wavelength. On the other hand, however, there were no such phenomenon observed for 59-P/M and 60-P/M. Similarly, the behaviour of self-assemblies to form stacked aggregates via Pt–Pt, π–π stacking, and dipole–dipole interactions is another effective method to enhance the dissymmetry glum factors. For example, two novel enantiomers of Pt(II) phenyl bipyridine complexes (61-R/S) with bis(phenylisoxazolyl) phenylacetylene ligands which were linked with a chiral alkyl chains were designed and synthesized. Both the complexes exhibited aggregation-induced helical assemblies in toluene, but nonhelical assemblies in chloroform. No appreciable CPL signals were observed either from the monomeric state of the molecule or in chloroform solution, whereas helical assemblies of 61-R/S showed mirror-image CPL spectra with a high glum up to 0.01 at the luminescence wavelength.
Additionally, two enantiomeric pure Pt(II) complexes (62-R/S) bearing the chromophoric tridentate ligand (2,6-bis(3-(trifluoromethyl)-1H- 1,2,4-triazol-5-yl)pyridine) and an ancillary ligand with a single asymmetric carbon, have no CD signals in tetrahydrofuran solutions and low luminescence efficiency of only 0.2%. Upon drop-casting tetrahydrofuran solution of 62-R/S onto a quartz substrate, the enantiomers exhibited CD signs of mirror images and high luminescence efficiency of 57% (62-R) and 37% (62-S), due to the formation of a long range–ordered chiral helical supramolecular configurations via Pt–Pt and π–π interactions.
Complexes with other ligands
With a similar strategy, two monomers (63-S and 64-S) where optically active (S)-3,7-dimethyloctyl groups are attached on both ends of the Pt(II) acetylide rod, were capable of forming supramolecular polymers by use of self-assembling π-aromatic monomer through intermolecular hydrogen bonds (63-S). Self-assembly of 63-S in a polar methylcyclohexane solution showed two maximum UV–vis absorption bands at 350 and 488 nm, which were assigned to intraligand π–π* (IL) transition and MLCT transition, respectively, and the emission locating at 566 nm resulted from MLCT. The bisignate CD signals were only observed for helical assemblies with long range order, which showed a moderate trisignate cotton effect for the IL transition and a weakly negative cotton effect (gabs = −0.00074) in the MLCT region, indicating the formation of single-handed polymeric assemblies. No CD signals could be detected under the same conditions for the counterpart monomer 64-S.
For more information on Pt(II) and other organometallic complexes, see ref. 29 and articles quoted therein.
The study by Schulte et al (ref. 45) is aimed at a heterobidentate trans-spanning coordinated compound, with C^N achiral binder, where the chiral centre is formed by
the central metal; the synthesis, carried in tetrahydrofuran from organic and metal-organic precursors, has resulted in a racemic mixture of two cyclometalated chiral complexes of Pt(II), one monodentate and one bidentate. The latter is present in both cis and trans form; this mixture has subsequently been separated into its chiral components by chromatographic means. Subsequently, enantiomers were characterized by 1H NMR spectroscopy and mass spectrometry, comparing some of
the results with DFT simulations of NMR spectra. An X-ray analysis was then carried out on crystallized samples of the three compounds, followed by the measurement of UV-vis and emission spectra, which were also supported by DFT calculations. Finally, the optical properties of circular dichroism and circularly polarized luminescence were analysed, in this case as well, with both experiments and TD-DFT simulations.
VIBRONIC SPECTROSCOPY
Internal coordinates
The majority of the models (both of the TI and the TD type) that are employed in the simulations of vibrationally resolved electronic spectra are based on the harmonic approximation: this is mainly a practical choice, one that has an impact on the definition of the vibrational wave function and generally on the truncation of the electronic transition dipole moment. This leads to far simpler equations than the anharmonic case, thus enabling faster computations and making vibronic spectroscopy applicable to many different systems; the best basis set to compute the integrals between vibrational wave functions (relying on second quantization) are the normal modes, which are obtained from the diagonalization of the Cartesian force constants matrix.
This approximation, however, work very well when applied to rigid and semi-rigid molecules. In the field of electronic spectroscopy, a system is semi-rigid or semi-semi-rigid if its structure remains unchanged or nearly unchanged after the electronic transition; more generally, a rigid system will have vibrations with high energy barriers, but the harmonic approximation may fail nonetheless in correctly representing them. The bounds will still dissociate if stretched too far, so the harmonic model gives a good approximation of the potential energy surface (PES) only close to the equilibrium. This is the main problem for electronic transitions, since the "vertical region" excitation (or de-excitation), can be shifted from the actual minimum. With a rigid system, the electronic transition is followed by a minimum change, so the minima of the 2 PES correspond to close structures.
If, on the other hand, the number of atoms and the flexibility of the system increases, and large-amplitude deformations accompany the electronic transitions, the harmonic picture of the process is no more accurate enough.
A possible solution to this problem consists in developing anharmonic models (based on variational or perturbative approaches) while maintaining the cartesian coordinate formalism; unfortunately, the computational cost of these methods tends to grow very quickly with increasing system size, to the point that their application is, in the majority of the cases, restricted to small-size molecules.
A simpler but nonetheless effective solution has its heart in the usage of a curvilinear set of internal coordinates. In the less elaborated case of purely vibrational spectroscopy it is possible to demonstrate that the same results can be obtained using either a cartesian or an internal type of coordinates (provided that the harmonic approximation still holds); on the other hand, when considering vibronic spectra, a single coordinate system is used to describe two different structures (that is, the equilibrium geometries of the initial and final states of the transition): most of the times, the shift between the geometries is not infinitesimal and depends on the coordinate set. The vibronic models developed in cartesian representation can be used in the internal one, provided that the quantity that changes with the coordinate system, the Duschinsky transformation, is appropriately formulated for a general set of curvilinear coordinates. It also possible (and it is preferred when dealing with large and flexible systems) to still work using normal coordinates, but express them in the internal coordinates bases: this last approach is the one adopted for the present work.
The majority of available vibronic models in internal coordinates are limited to the time-independent simulation of one-photon absorption spectra at the Franck-Condon level for small and medium sized systems. A few years ago, the authors of ref. 3 extended a previously developed computational platform, designed to evaluate vibronic effects, both in time-independent and time-dependent formulation, in various kind of electronic spectroscopies (one-photon emission and absorption, conventional and chiral), to support internal coordinates. First and second derivatives of different sets of internal coordinates with respect to cartesian ones have been implemented, thus enabling the usage for both vertical and adiabatic models, including also mode mixing and Herzberg-Teller contributions. Furthermore, the building of all the supported nonredundant sets of internal coordinates is completely automized: the algorithm employs a primitive redundant set that is derived from molecular topology and an extended one supporting special types of coordinates (like ring puckering). Thanks to the presence many different types of coordinates (together with the traditional stretching, bending and torsional, also linear and out-of-plane sets are
included), specific systems, such as inter-molecular hydrogen bonding, can be properly treated.
After the analysis of a series of case studies (organic molecules of varying complexity), the authors concluded that cartesian and internal coordinates can be considered nearly equivalent for semi-rigid systems that do not exhibit significant geometric distortions between the initial and final electronic states. On the other hand, when studying molecular systems that are more flexible or show significant structural deformations caused by the transition, delocalized (and possibly weighted) internal coordinates, based on generalized internal coordinates (GICs) are of much more effectiveness than cartesian ones.
Flexible molecules
While the usage of curvilinear internal coordinates notably improves the evaluation of the vibronic effects at the harmonic level (as they reduce the coupling between modes), the latter remains insufficient in the case of predominance of large amplitude motions (LAMs), which can also occur in medium-sized systems.
The authors of ref. 4 proposed a method that is specifically built to target the flexibility problem in the case of medium- to large-size molecules. The method is essentially a hybrid approach, in which the single degree of freedom associated to the large-amplitude motion accompanying the electronic transition is decoupled from the other modes (treated at the harmonic level) and described in a full-variational anharmonic way. An internal coordinates formalism is employed to sensibly reduce the coupling between the LAM and the other harmonic degrees of freedom; the PESs of the initial and final states along the LAM are computed by means of a scan calculation, while the transition properties between the vibrational levels of the two states involved are obtained with a pseudo-variational approach, relying on a basis set which derives from the variational basis set representation (VBR) but does not require an analytic form of the operators (wave functions, hamiltonian). Arbitrarily complex deformations are supported, thanks to the freedom in the choice of the path hamiltonian model (either reaction or internal coordinates) for the definition of the coordinate associated to the LAM. Unlike many of the other models available, the variations of the harmonic normal modes along the LAM are appropriately considered within an adiabatic picture. The description of the harmonic degrees of freedom is based on the models developed for fully harmonic simulations, thus supporting both adiabatic and vertical models with the possible inclusion of mode-mixing and Herzberg-Teller effects. The model was applied to several test cases: the results show that this hybrid scheme can be applied not only to standard deformations (like inversions and torsions along a dihedral angle) but also to more complex,
large-amplitude motions, such as ring deformations. An interesting feature is the possibility to simulate effectively a whole class of spectra with just a small increase of the computational cost.
Phosphorescent materials
Phosphorescent materials, particularly those containing transition metal complexes, can be employed in many different fields and have recently been the object of extensive studies, thanks in part to their applicability in electroluminescent devices such as organic light-emitting diodes, sensors, and probes. As already mentioned, emitting systems which exhibit chirality are particularly appealing because of their possible application in data storage, directional backlight 3D and liquid crystal displays, as spin sources in optical spintronics and information carriers in quantum computing. Phosphorescence spectroscopy is the main method for studying phosphorescent materials; however, the characterization of chiral systems requires a chiral spectroscopy, such as circularly polarized luminescence; when associated to triplet-singlet transitions, this technique is also known as circularly polarized phosphorescence (CPP). Although less common, in particular because of their high sensitivity and low signal strengths, circularly polarized luminescence and CPP are emerging as powerful techniques to complement electronic circular dichroism and provide more extensive characterizations of excited electronic states.
As detailed in ref. 22, Egidi et al. have developed a method to compute both phosphorescence and CPP spectra that are resolved at the vibrational level, including both Franck-Condon and Herzberg-Teller effects. The harmonic approximation is used when describing the singlet and triplet states involved in the transition and the methods include mode mixing effects in computing the Franck-Condon integrals. In order to take into account the effects of spin-orbit couplings, a relativistic two-component time-dependent density functional theory method is adopted. The method was applied to a relatively small and rigid organic molecule and to various transition metal complexes, with generally satisfying results.
THEORETICAL METHODS
DENSITY FUNCTIONAL THEORY (DFT)
A powerful computational tool that allows to simulate the excited state properties of molecular systems is given by the theory of functional density dependent on time (Time-dependent density functional theory, TD-DFT): dating back to the 80’s of the last century (the Runge-Gross theorem, TD-DFT’s cornerstone, dates from 1984), this theory is based on the same conceptual assumptions as the previously known DFT to investigate the dynamic properties of systems to many bodies subject to potential variables over time (such as electric or magnetic fields). The effects of these fields on molecular and solid-state systems allow to use TD-DFT to calculate excitation energies, photoabsorption or emission spectra and other response properties.
Density functional theory is based on a completely different approach to traditional methods used in condensed matter physics and quantum chemistry, such as Hartree-Fock. The fundamental physical quantity of these “classic” methods (next to the Hartree-Fock, the configuration interaction and the coupled cluster can also be found) is the wave function of the many body system (these methods are therefore called wave function methods), which depends, for a system composed of N electrons, on 3N spatial variables. Calculating the exact wave function of a solid, containing a number of atoms of the order of 1023, is obviously an insurmountable problem;
considerable difficulties arise already at the time of processing systems of the size of a medium- or large-sized molecule.
The DFT approach, developed in its original form by Hohenberg and Kohn in 1964 and then formalized by Kohn and Sham the following years, aims at simplifying computational analysis by replacing the wave function with the electronic charge density, which depends only on three spatial variables, regardless of the number of electrons actually present in the system.
In their seminal 1964–1965 papers, Hohenberg, Kohn, and Sham founded the rigorous theory that finally legitimized the intuitive leaps of Thomas, Fermi, Dirac, and Slater. Thus, 1964 is widely accepted as the birth year of modern DFT.
There are methods that return system properties with better accuracy than (TD)DFT; however, as stated above, due to the high computational cost, they are only applicable to very small systems, of the order of a few atoms or little more, making
(TD)DFT the best methodology to deal with not too small molecules and solid state systems.
Although from a strictly formal point of view (TD)DFT is a formulation free from approximation of the problem to many bodies, in practice some form of inaccuracy is inevitable, because the precise form of the functional exchange-correlation is unknown (often we speak in fact of TD-DFA: Time-Dependent Density Functional Approximation). Over the years, many different functions have been proposed (LDA: Local density Approximation; GGA: Generalized Gradient Approximation; functional hybrids; etc.), applicable with greater or lesser reliability to different types of systems. In particular, in recent years there has been a proliferation of functions, many of which have been heavily parameterized (often based on organic systems), to the point that they are applicable only to a small number of cases, often close to the original set used for their definition. For this reason, if you want to use (TD)DFT to study a new system, you need to perform a benchmark to make a rational choice of the functional and base (see below).
The most commonly used functionals in the study of organometallic compounds are the above-mentioned hybrid ones, introduced by Becke in 1993 to improve the performance of DFT in the calculation of properties such as atomization energy, binding lengths and vibrational frequencies: this particular functional class is distinguished by the incorporation of a portion of “exact exchange”, derived from the Hartree-Fock theory, in the rest of the energy exchange-correlation, which comes from other approximations such as LDA and GGA; in the expression of the functional hybrids appear different parameters, often, but not always, of empirical origin. Hybrid functionals worthy of mention are B3LYP and PBE0.
A benchmarking process is usually required to select the functional and basis set appropriate to the system to be analysed, which consists in comparing the results obtained with regard to a system which is as similar as possible to the one under study using various functionalities with bases of different types. Benchmarking can be done against experiments or theory: while in the vast majority of cases the greatest possible adherence to empirical data is desirable, it is often very complicated (and virtually impossible for a large number of molecules in solution) to perform a simulation taking into account the whole experimental set-up and often the correspondence between the simulated and measured properties is not so apparent. On the other hand, a comparison with very accurate wave function theories allows quick and direct comparisons, but these are limited by the availability of theoretical data (mostly restricted to small molecules).
It was established in the 1964 paper of Hohenberg and Kohn that the total electron density ρ completely and exactly determines all the ground-state properties
of an N-electron system. Thus, ρ can be used as the fundamental “variable” in electronic structure theory. The much more complicated N-electron wave function is, in principle, unneeded.
For a system of N interacting electrons in an external potential, vext ,there is a
unique ground-state wave function Ψ0 and associated density ρ.
𝑣𝑒𝑥𝑡 → 𝛹0∨ 𝑣𝑒𝑥𝑡 → 𝜌
If the mapping from vext to ρ is “one-to-one” or reversible, then
𝜌 → 𝑣𝑒𝑥𝑡 → 𝛹0
i.e., ρ uniquely determines vext , which has a unique Ψ0, and thus in principle everything
is known about the system.
The above is not enough, however. For the theory to be self-contained, a variational principle is needed. Corresponding to the first and last terms in the Hamiltonian, the terms that do not involve the external potential, there exists a density functional for the total kinetic and total Coulomb interaction energy:
𝐹(𝜌) = 𝑇(𝜌) + 𝑉𝑒𝑒(𝜌)
From the WFT variational principle, it can be proved that:
𝐹(𝜌′) + ∫ 𝑣
𝑒𝑥𝑡𝜌′ ≥ 𝐹(𝜌) + ∫ 𝑣𝑒𝑥𝑡𝜌 = 𝐸0
where ρ’ is not the ρ corresponding to vext, but to some other external potential, and
E0 is the exact ground-state energy. This is the Hohenberg-Kohn density variational
principle.
It is noteworthy that the Hohenberg-Kohn proofs are restricted to nondegenerate ground states, and that the density-potential mapping inherently
assumes “v-representable” densities, a subset of all conceivable Fermion densities. The later “constrained search” approach of Levy relaxes this requirement.
That the functionals T(ρ) and Vee(ρ) are known to exist does not imply that their expression is known. The requirement of v-representability (or the looser Fermion representability in the approach of Levy) on the densities is an additional difficulty. Nonrepresentable variational densities will collapse to the same shell-structureless densities obtained in Thomas-Fermi-Dirac theory. A year later, Kohn and Sham addressed both of these problems.
Considered a single Slater determinant of orthonormal orbitals 𝜓I, the total density is:
𝜌 = 2 ∑|𝜓𝑖|2 𝑖
and the total kinetic energy:
𝑇0 = −1
2∑ 2 ∫ 𝜓𝑖
∗𝛻2 𝑖
𝜓𝑖
where, for simplicity spin-neutral systems are assumed.
A single Slater determinant connotes independent, noninteracting electrons. Nevertheless, we presume that the density expression spans all possible N-electron densities, interacting or not. It is reasonable to assume that T0 is a rather good
approximation to T(ρ) in the Hohenberg-Kohn energy functional F(ρ). It is also reasonable to approximate Vee(ρ) by the classical Coulomb self-energy
𝐽(𝜌) =1 2∫ ∫
𝜌(1)𝜌(2) 𝑟12 𝑑1𝑑2
Kohn and Sham called the error made by these approximations the exchange-correlation energy EXC:
𝐸𝑋𝐶(𝜌) = 𝑇(𝜌) + 𝑉𝑒𝑒(𝜌) − 𝑇0(𝜌) − 𝐽(𝜌)
where T0, by extension of the Hohenberg-Kohn analysis to noninteracting systems, is
a density functional also, and therefore so is EXC. EXC is composed of both kinetic and
potential energy terms.
Collecting all the aforementioned quantities, the Kohn-Sham total energy functional is:
𝐸(𝜌) = 𝑇0(𝜌) + ∫ 𝜌𝑣𝑒𝑥𝑡 + 𝐽(𝜌) + 𝐸𝑋𝐶(𝜌)
The brilliance of this decomposition is that T0 and J are given by exact
expressions, and that the “unknown” functional, EXC, is a relatively small part of the total.
Variational minimization of the previous equation with respect to the orbitals 𝜓i yields the Kohn-Sham (KS) orbital equation:
−1 2∇
2𝜓
𝑖 + 𝑣𝐾𝑆𝜓𝑖 = 𝜀𝑖𝜓𝑖
where vKS has the expression:
𝑣𝐾𝑆 = 𝑣𝑒𝑥𝑡+ 𝑣𝑒𝑙+𝛿𝐸𝑋𝐶 𝛿𝜌
and δEXC/δρ is the functional derivative of EXC with respect to ρ. The theory is now
complete. Electrons in atoms, molecules, and solids can be viewed as independent particles, moving in the effective potential vKS.
Kohn-Sham DFT is operationally an independent-particle theory, simpler even than Hartree-Fock, but it delivers, in principle, the exact density and exact total energy any interacting, correlated electronic system. Everything depends on the functional
EXC(ρ) and its functional derivative, whose existence is certain, but for which no explicit
expression is known. In 1965, the quest for the holy grail of electronic structure theory began.
Kohn and Sham proposed a simple model for EXC, the so-called “local density approximation” (LDA):
𝐸𝑋𝐶𝐿𝐷𝐴= ∫ 𝑒𝑋𝐶𝑈𝐸𝐺
where eUEG
XC (ρ) is the exchange-correlation energy, per unit volume, of a uniform
electron gas having the local value ρ(r) of the density. It is a reasonable first approximation, in the same vein as Thomas-Fermi-Dirac theory, certainly good for a very slowly varying ρ(r). It is surprisingly good for atoms and molecular clusters, too.
It was thought that the local density gradient could provide useful additional information. Dimensional analysis fixes the form of the lowest-order gradient correction (LGC) to the LDA as follows:
𝐸𝑋𝐿𝐺𝐶 = 𝐸
𝑋𝐿𝐷𝐴− 𝛽 ∑ ∫
(∇𝜌𝜎)2
𝜌𝜎4/3
𝜎
with a coefficient β, one might hope to calculate from the theory of the slowly varying electron gas. In the 1970s and 80s, at least three different theoretical values were published.
Earlier, in 1969, the pragmatic approach of fitting β to atomic exchange energies had been taken by Herman et al. The Herman value, β = 0.003 to 0.004, is at least twice as large as the later “theoretical” values, a conundrum that persists to this day. There is no doubt, however, that the “theoretical” β is of little relevance in chemistry. Becke has published a simple exchange hole model giving roughly the Herman value.
There is a problem with the equation that renders it unacceptable in any case. The functional derivative (i.e., the Kohn-Sham potential) of the gradient term diverges in the exponential tails of finite systems. This is not just a computational nuisance. It reflects a fundamental failure of the LGC functional form. Becke suggested a simple modification in 1986 that fixes the problem:
𝐸𝑋𝐵86 = 𝐸𝑋𝐿𝐷𝐴− 0,0036 ∑ ∫ 𝜌 𝜎 4/3 𝜒𝜎2 1 + 0,004𝜒𝜎2 𝜎 𝜒𝜎 = (∇𝜌𝜎)2 4/3
known as “B86,” with parameters fit to atomic data. The B86 “β” (0.0036) is in good agreement with the Herman value. Particularly exciting, however, was that B86, in combination with an opposite-spins-only LDA correlation functional, gave excellent bond energies in diatomic molecules.
Becke published a second gradient-corrected EX the same year (“B86b”):
𝐸𝑋𝐵86𝑏 = 𝐸𝑋𝐿𝐷𝐴− 0,00375 ∑ ∫ 𝜌𝜎4/3 𝜒𝜎
2
(1 + 0,007𝜒𝜎2)4/5 𝜎
The performance of B86b+PBE is similar to that of B86+PBE in predicting atomization energies.
At the same time, Perdew and Wang took another approach. The exchange hole always has negative value and is always normalized to −1 electron. Upon deriving an expression for the exchange hole in an electron gas with a small density gradient, they found that this lowest-order (in the density gradient) hole was not negative definite and did not satisfy the normalization condition. They put equal to zero the positive regions, restored proper normalization by truncating the rest, and numerically integrated the truncated hole to obtain an EX approximation. Then the numerical results were fit with the following functional form:
𝐸𝑋𝑃𝑊86(𝜌) = ∫ 𝑒𝑋𝐿𝐷𝐴(1 + 2,96𝑠2+ 14𝑠4+ 0,2𝑠6)1/15 𝑒𝑋𝐿𝐷𝐴 = −3 4( 3 𝜋) 1 3 𝜌4/3 , 𝑠 = |∇𝜌| 2(3𝜋2)1/3𝜌4/3
known as “PW86.” Perdew and co-workers, as is the convention of physicists, write their exchange functionals as spin neutral total-density functionals.
Without any parameters fit to atomic data, PW86 atomic exchange energies compare well to those of B86 and B86b.
Exchange functionals of both density and density gradient can be expressed with the following formula:
𝐸𝑋(𝜌) = ∫ 𝑒𝑋𝐿𝐷𝐴(𝜌)𝑓 𝑋(𝑠)
where the variable s is a dimensionless density gradient and fX(s) is the so-called
exchange enhancement factor; this factor in an even function that can be expanded in powers of s2; moreover, fX(0) = 1 (to satisfy the uniform electron gas limit); PW86, B86 and B86b can all be written in this form, that is known as the Generalized Gradient Approximation (GGA).
In 1993, Becke observed that GGAs, while dramatically reducing the massive overbinding tendency of the LDA, show a small overbinding tendency still. This can be understood from the adiabatic connection formulas: EXC depends on the
coupling-strength averaged exchange-correlation hole. The λ = 0 exact exchange hole is relatively delocalized (“nonlocal”) in multicentre systems; GGA holes, however, are inherently localized for every λ, including λ = 0, and are for this reason too compact. This explains why exchange-correlation GGAs, despite their sophistication, are slightly overbinding. The delocalized character of the exchange-correlation hole at λ = 0 can only be captured by replacing a small amount of DFA exchange by exact exchange.
In 1993 the following replacement was proposed by Becke:
𝐸𝑋𝐶𝐵3𝑃𝑊91 = 𝐸𝑋𝐶𝐿𝐷𝐴+ 𝑎(𝐸𝑋𝑒𝑥𝑎𝑐𝑡− 𝐸𝑋𝐿𝐷𝐴) + 𝑏∆𝐸𝑋𝐵88+ 𝑐𝛥𝐸 𝐶𝑃𝑊91
This preserves the uniform electron gas limit but reduces the amount of the B88 gradient correction ΔEB88
X and the PW91 gradient correction ΔEPW91C because
substitution of exact exchange reduces their importance. The three parameters a, b, and c were fit to atomization-energy data. This functional is known as “B3PW91” in deference to its three fitted parameters.
Frisch and co-workers reworked B3PW91 using the LYP67 correlation DFA instead of PW91. The reworked functional employs the same three parameters as in the previous equation and is known as “B3LYP.” For the past two decades, B3LYP has been the most popular exchange-correlation DFA in computational chemistry.
Perdew, Ernzerhof and Burke argued that 25% exact exchange was preferable to the 20% in B3PW91. Combined with the PBE exchange-correlation GGA, and fixing
b = 0.75 and c = 1, the corresponding functional is called “PBE0”. The slightly higher
exact-exchange fraction, a = 0.25, in PBE0 compensates for the distinctive overbinding tendency of the pure PBE+PBE GGA.
These functionals are called “hybrid” functionals for a clear reason, as they mix GGA exchange with explicitly nonlocal exact exchange. Also, their implementation in the GAUSSIAN program, and in other programs, was a hybrid of technologies. The exact exchange part was implemented with well-developed Hartree-Fock techniques
(differentiation with respect to orbital expansion coefficients) and the rest was KS-DFT: as such, the EX
exact part is not Kohn-Sham exact exchange. The orbitals are nevertheless close to true Kohn-Sham orbitals. Mixing of BR-type functionals and meta-GGAs with exact exchange has also been widely tested.
For further details on DFT history, see ref. 10 and references quoted therein.
POLARIZABLE CONTINUUM MODEL (PCM)
A problem to solve in order to simulate the behaviour of a molecule in solution is that of the proper modelling of the solute-solvent interactions, which almost always influence the properties of the molecule itself. If each solvent molecule were considered separately, the computational cost of calculation would be prohibitive; a possible solution is to adopt the polarizable continuum model (PCM: Polarizable Continuum Model, dating back to 1981), according to which discrete solvent molecules are replaced by a continuous and polarizable medium (dielectric or conductive, depending on the system being analysed). This makes it easier to perform simulations ab initio.
The model consists of a molecular system (the solute) put inside a void cavity surrounded by a continuous medium that approximates the solvent; the cavity (whose shape should in theory reproduce as well as possible the molecular shape, but in practice is often spherical or elliptical for ease of computation) should exclude the solvent and include as much as possible of the solute charge distribution. These requirements not completely fulfillable, and a certain charge density overlap with the solvent medium is always present.
Because of the irregular shape of most molecules, there are small portions of space at their borders that are not accessible to the solvent. For this reason, alongside the van der Waals volume, the concepts of solvent-excluding and solvent accessible surface (SES and SAS, respectively) are introduced: in both cases, the solvent molecule is approximated with a sphere, whose volume is equal to the van der Waals one. The positions assumed by the centre of a solvent sphere rolling on the van der Waals surface of the solute define the SA surface, that is, the surface enclosing the volume in which the solvent centre cannot enter. The same sphere used as a contact probe on the solute surface defines the SE surface, that is, the surface enclosing the volume in which the whole solvent molecule cannot penetrate (see figure 6).
The electrostatic problem of the evaluation of the interaction energy between solute and solvent, including also mutual polarization effects, is solved by introducing an apparent charge distribution s spread on the cavity surface. In the computational practice this continuous distribution is discretized by point charges qi, each associated with a small portion (tessera) of the cavity surface, and defined through a set of linear equations written in the following matrix formulation:
𝒒 = 𝜮 = −𝜮𝑫−𝟏𝑬𝒏
Here S is a diagonal square matrix with elements given by the areas of the surface tesserae and En is the column vector containing the normal components of
the electric field due to the solute. D is a nonsymmetric square matrix with dimension equal to the number of tesserae, whose elements depend on geometrical cavity parameters and on the dielectric constant.
Figure 6Solvent accessible surface (SAS) traced out by the centre of the probe representing a solvent molecule. The solvent excluded surface (SES) is the topological boundary of the union of all possible probes that do not overlap with the molecule (from ref. 48).
The procedures for the calculation of solute energy and wave function with the PCM method have been reported in several papers. Suffice it to say that, in order to get solvation quantities, one has to resort to a direct minimization of the functional of the free energy G of the whole solute–solvent system.
For a closed-shell solute, described by an SCF wave function with orbitals expanded over a finite basis set, the variational condition δG=0 leads to the following equation:
𝑭̃𝑪 = 𝑺𝑪𝜺
where the tilde emphasizes that the Fock matrix contains terms accounting for the presence of the solvent field; namely:
𝑭̃ = 𝒉 +1
2(𝒋 + 𝒚) + 𝑮(𝑷) + 𝑿(𝑷)
Here h and G(P) collect the usual one- and two-electron integrals over the basis set used for the vacuum SCF calculation, while P is the one-electron density matrix; the matrices j, y and X(P) collect the one- and two-electron integrals to be added when the interactions with the polarized dielectric medium are explicitly taken into account within the PCM framework.
The analysis starts by considering two charge distributions, ρ and ρ’, carried by the solute molecule and both located inside the cavity Ωi. Their electrostatic interaction energy is given by:
𝐸𝐼(𝜌, 𝜌′) = ∫ 𝜌′(𝑥)𝑉(𝑥)𝑑𝑥
where the electrostatic potential V created by r satisfies the Poisson equation:
−∇ ∙ (𝜺(𝑥) ∗ ∇𝑉(𝑥)) = 𝜌(𝑥)
𝜺(𝒙) = {𝜺𝒊 𝑖𝑓 𝑥 ∈ 𝛺𝑖 𝜺𝒆 𝑖𝑓 𝑥 ∈ 𝛺𝑒
where εi is the unit 3*3 tensor, εe a positive 3*3 symmetric tensor, and Ωe the outer
domain. Without loss of generality, a set of orthonormal coordinates of the real space so that εe is diagonal can always be chosen.
Denoting by Gi and Ge the Green’s functions of the operators -Δ and -div(ε gradV), respectively, the following function is defined:
𝐺(𝑥, 𝑦) = { 𝐺𝑖(𝑥, 𝑦) = 1 4𝜋|𝑥 − 𝑦| 𝑖𝑓 𝑥 ∈ 𝛺𝑖 𝐺𝑒(𝑥, 𝑦) = 1 4𝜋√𝑑𝑒𝑡𝜺𝒆(𝜺𝒆−𝟏(𝑥 − 𝑦)) ∗ (𝒙 − 𝒚) 𝑖𝑓 𝑥 ∈ 𝛺𝑒
By introducing the electrostatic potential ϕ’(x) generated by the distribution ρ’ in the vacuum, and a function f(x) such as:
𝜙′(𝑥) = ∫ 𝐺𝑖(𝑥, 𝑦)𝜌′(𝑦)𝑑𝑦
𝑓(𝑥) = ∫ 𝐺(𝑥, 𝑦)𝜌(𝑦)𝑑𝑦
it may be defined an ‘‘apparent’’ potential W=V-f , and rewritten the interaction energy of as:
𝐸𝐼(𝜌, 𝜌′) = 𝐸1+ 𝐸2 = ∫ 𝜌′(𝑥)𝑓(𝑥)𝑑𝑥 + ∫ 𝜌′(𝑥)𝑊(𝑥)𝑑𝑥
The first term E1 is easy to compute, since that both charge distributions are
supported in Vi: this is in fact the expression of the electrostatic energy in the vacuum. The point is to compute the second term: it can be shown that it can be rewritten by
introducing the quantity σ = S-1i*Wi, which has the dimension of a surface charge, so
that:
𝐸2 = ∫ 𝜎(𝑥)𝜙′(𝑥)𝑑𝑥
The problem is thus completely solved if we manage to compute the equivalent surface charge σ.
It is worth remarking that, when εe is a scalar i.e., when the dielectric is isotropic,
we obtain (𝜀𝑒+ 1 𝜀𝑒− 1 𝐼 2− 𝐷𝑖 ∗) ∗ 𝜎 = −𝐸 𝑖
which is exactly the operator-like form of the matrix equation at the beginning of this paragraph, obtained for the standard PCM procedure. This is a very important point as it clearly shows that the new integral formulation approach reduces to that of PCM when the anisotropy of the medium disappears.
As a concluding remark, it is worth saying that, in spite of its successes, PCM is unable to represent specific solute-solvent interactions and in this case, explicit molecules may need to be included in the cavity. The actual solute becomes the original solute with the addition of a few solvent molecules.
BASIS SETS AND PSEUDOPOTENTIAL
In addition to choosing the appropriate exchange-correlation function, another problem to be solved for analysing a system with (TD)DFT is selecting the right base of functions.
A basis set in is a set of functions that is employed to expand the electronic wave function in theoretical chemistry models (such as Hartree-Fock or DFT), in order to transform the differential equations into algebraic equations easier to implement efficiently in a computational software.
For chemical calculations, basis sets are usually composed of atomic orbitals of the Slater or Gaussian type (STO and GTO respectively, the latter being by far the most used); plane waves are instead preferred by physicists. STOs are solutions to
