Hund's multiplicity rule: from atoms to quantum dots

In 1965, Davidson has shown that the textbook explanation for the Hund's multiplicity rule in atoms, based on the Pauli principle, is wrong. The reason for the failure of the textbook proof, as has been given later by others and as appears today in modern textbooks, it is based on the need to introduce angular electronic correlation into the calculations. Here, we investigate an applicability of this argumentation for helium and for the case of two-electron spherically symmetric rectangular quantum dots (QDs). We show that, for helium and also for the QD, the differences between the singlet and triplet excited states can be explored by calculations within the framework of the mean-field approximation, and, surprisingly, without the need of introducing the angular electronic correlation. Moreover, our calculations have shown that the triplet state of the QD is lower in energy than the corresponding singlet state due to lower electronic repulsion contribution, exactly as being assumed in the oldest explanation of the Hund's rule based on the Pauli principle.     

Violation of Hund's rule in the lowest excited singlet-triplet pairs of dicyclohepta[cd,gh]pentalene and dicyclopenta[ef,kl]heptalene

Using the Pariser-Parr-Pople type MO method, the energy ordering of the lowest excited singlet-triplet pairs of the title molecules is examined by taking into account ample configuration interactions. In dicyclohepta[cd,gh]pentalene, it is shown that the lowest excited singlet state lies below the corresponding triplet state by about 6 kcal/mol. This violation of Hund's rule is ascribed entirely to the correlation effects brought about by spin polarization terms. Such a violation of Hund's rule is expected to occur in the lowest excited states of dicyclopenta[ef,kl]heptalene.     

The influence of correlation on the interpretation of Hund's multiplicity rule: A quantum Monte Carlo study

A systematic quantum Monte Carlo study of 2p atoms (C, N, O) and 3p atoms (Si, P, S) is performed to investigate the influence of correlation on the interpretation of Hund's multiplicity rule, which is an extension of our previous study of the carbon atom [J. Chem. Phys. 121, 7144 (2004)] to heavier atoms. The accuracy in the present study is significantly improved as compared with the previous study. A detailed analysis of the correlation contribution to individual energy components of the total energy is given beyond the self-consistent Hartree-Fock calculation. The stability of the highest spin-multiplicity state of all the atoms is ascribed to the greater electron-nucleus attraction energy that is gained at the cost of increasing the electron-electron repulsion energy as well as the kinetic energy. The present study demonstrates that correlation does not change the above conclusion due to the Hartree-Fock theory to support Boyd's less screening mechanism.     

Influence of electron correlation on the character of the splitting of degenerate states in metalloporphyrin dianions

The lowest singlet and triplet energy levels of metalloporphyrin dianions have been calculated according to the extended Hartree-Fock method in the -electronic approximation with consideration of electron correlation. The calculations confirm the deviation from Hund's rule observed for these compounds. The possibility of the application of the method indicated to the construction of many-electron functions with the correct symmetry in the case of orbital degeneracy has been discussed.Translated from Teoreticheskava i Éksperimental'naya Khimiya, Vol. 24, No. 2, pp. 208–211, March–April, 1988.     

Second-order correlation potential in the Kohn–Sham approximation for atoms

A new type of correlation functional derived from the second-order expression for the correlation energy of an atom is proposed. The derived correlation potential contains one free parameter, which is determined by fitting the known pair correlation energy. The calculations with this potential in the Kohn–Sham approximation give rather accurate values for the matrix elements of different operators.     

Unified interpretation of Hund's first and second rules for 2p and 3p atoms

A unified interpretation of Hund's first and second rules for 2p (C, N, O) and 3p (Si, P, S) atoms is given by Hartree-Fock (HF) and multiconfiguration Hartree-Fock (MCHF) methods. Both methods exactly satisfy the virial theorem, in principle, which enables one to analyze individual components of the total energy E(=T+V(en)+V(ee)), where T, V(en), and V(ee) are the kinetic, the electron-nucleus attraction, and the electron-electron repulsion energies, respectively. The correct interpretation for each of the two rules can only be achieved under the condition of the virial theorem 2T+V=0 by investigating how V(en) and V(ee) interplay to attain the lower total potential energy V(=V(en)+V(ee)). The stabilization of the more stable states for all the 2p and 3p atoms is ascribed to a greater V(en) that is caused by contraction of the valence orbitals accompanied with slight expansion of the core orbitals. The contraction of the valence orbitals for the two rules is a consequence of reducing the Hartree screening of the nucleus at short interelectronic distances. The reduced screening in the first rule is due to a greater amount of Fermi hole contributions in the state with the highest total spin-angular momentum S. The reduced screening in the second rule is due to the fact that two valence electrons are more likely to be on opposite sides of the nucleus in the state with the highest total orbital-angular momentum L. For each of the two rules, the inclusion of correlation does not qualitatively change the HF interpretation, but HF overestimates the energy difference ∣ΔE∣ between two levels being compared. The magnitude of the correlation energy is significantly larger for the lower L states than for the higher L states since two valence electrons in the lower L states are less likely to be on opposite sides of the nucleus. The MCHF evaluation of ∣ΔE∣ is in excellent agreement with experiment. The present HF and MCHF calculations demonstrate the above statements that were originally given by Katriel [Theor. Chem. Acta 23, 309 (1972); 26, 163 (1972)]. We have, for the first time, analyzed the correlation-induced changes in the radial density distribution for the excited LS terms of the 2p and 3p atoms as well as for the ground LS term.     

Iridium cyclometalated complexes with axial symmetry: time-dependent density functional theory investigation of trans-bis-cyclometalated complexes containing the tridentate ligand 2,6-diphenylpyridine

A new series of iridium cyclometalated complexes with a C/N/C dppy-type ligand and a N/N/N tpy-type ligand have been synthesized and characterized by various techniques such as mass spectrometry, 1H and 13C NMR, cyclic voltammetry, both steady-state and time-resolved emission and absorption studies, and time-dependent DFT (TDDFT) calculations. The complexes exhibit strong visible absorptions and long-lived (1.6-2.0 micros) emissions (lambdamax, ca. 680 nm) in room-temperature solution. DFT calculations on the ground-state geometry match that of an X-ray crystal structure. TDDFT calculations give accurate predictions of the electronic absorption energies and intensities, while geometry optimizations on the lowest energy triplet state give accurate energies for the emission. Examination of the relevant molecular orbitals shows that the inherent asymmetry of the coordination environment offers a unique directional character to the emitting excited state, which is predominately LLCT (dppy --> tpy) in nature.     

Modeling the adiabatic connection in H2

Full configuration interaction (FCI) data are used to quantify the accuracy of approximate adiabatic connection (AC) forms in describing the ground state potential energy curve of H2, within spin-restricted density functional theory (DFT). For each internuclear separation R, accurate properties of the AC are determined from large basis set FCI calculations. The parameters in the approximate AC form are then determined so as to reproduce these FCI values exactly, yielding an exchange-correlation energy expressed entirely in terms of FCI-derived quantities. This is combined with other FCI-derived energy components to give the total electronic energy; comparison with the FCI energy quantifies the accuracy of the AC form. Initial calculations focus on a [1/1]-Padé-based form. The potential energy curve determined using the procedure is a notable improvement over those from existing DFT functionals. The accuracy near equilibrium is quantified by calculating the bond length and vibrational wave numbers; errors in the latter are below 0.5%. The molecule dissociates correctly, which can be traced to the use of virtual orbital eigenvalues in the slope in the noninteracting limit, capturing static correlation. At intermediate R, the potential energy curve exhibits an unphysical barrier, similar to that noted previously using the random phase approximation. Alternative forms of the AC are also considered, paying attention to size extensivity and the behavior in the strong-interaction limit; none provide an accurate potential energy curve for all R, although good accuracy can be achieved near equilibrium. The study demonstrates how data from correlated ab initio calculations can provide valuable information about AC forms and highlight areas where further theoretical progress is required.     

Rotational energy transfer of SH(X2Π, v'=0, J'=0.5-10.5) by collision with Ar: Λ-doublet resolved transition propensity

The behavior of Λ-doublet resolved rotational energy transfer (RET) by Ar collisions within the SH(X(2)Π, v'=0) state is characterized. The matrix elements of terms in the interaction potential responsible for interference effects are calculated to explain the propensity rules for collision-induced transitions within and between spin-orbit manifolds. In this manner, the physical mechanisms responsible for the F(1)-F(1), F(2)-F(2), and F(1)-F(2) transitions may be reasonably identified. As collision energy increases, the propensity for collisional population of the final e or f level is replaced by the e/f-conserving propensity. Such a change in propensity rule can be predicted in terms of energy sudden approximation at high J limit for the pure Hund's case scheme.     

An examination of density functional theories on isomerization energy calculations of organic molecules

Long-range corrected (LC) density functional theories (DFTs) were applied to the isomerization energy calculations of organic molecules to make clear why conventional DFTs including B3LYP have given poor isomerization reaction energies. Combining with local response dispersion (LRD) method, we performed LC-DFT calculations for the benchmark set of isomerization reactions. Consequently, we found that LC-DFT?+?LRD methods give accurate reaction energies equivalent to up-to-date DFTs containing many semi-empirical parameters. This result indicates that long-range exchange and intramolecular dispersion correlation interactions, which have been neglected in conventional DFTs, play prominent roles in isomerization reactions. However, we also found that these interactions are not sufficient to give accurate isomerization energies especially for cyclization reactions. Considering that Gaussian-attenuated LC-DFTs (LCgau-DFTs) give better isomerization reaction energies than LC-DFTs, we suggested that the isomerization energies will be further improved by correcting the short-range part of exchange functionals in DFT with keeping the whole long-range exchange interactions.     

With a little help from my friends: forty years of fruitful chemical collaborations

Over the past 40 years, much of the author's research, both computational and experimental, has involved collaborations. This Perspective describes some of the author's collaborative research in eight different areas of organic and theoretical chemistry: (1) hydrocarbons containing unsaturatively, 1,3-bridged cyclobutane rings, (2) the use of orbital topology for predicting the ground states of diradicals, (3) violations of Hund's rule, (4) the chemistry of phenylnitrenes, (5) tunneling by carbon in organic reactions, (6) the Cope rearrangement and the effect of substituents on it, (7) pyramidalized alkenes, dehydrocubanes, cubyl cation, and octanitrocubane, and (8) the effects of geminal fluorine substitution at C-2 of 1,3-diradicals. Highlighted in this Perspective are the synergism between calculations and experiments in the author's research and the many different roles that serendipity has played in the collaborations that are described herein.     

Planar tetracoordinate carbon species involving beryllium substituents

Planar tetracoordinate carbon (ptC) arrangements can be achieved by employing multiple substituents based on beryllium, despite its rather weak pi-acceptor ability. A variety of ptC-containing examples, some with more than one ptC, have been designed computationally by elaborating the planar C(BeH) 4 (2-) prototype at B3LYP/6-311++G(3df,2p) and MP2/6-311++G(3df,2p) levels of theory for some small ptC representatives. The prototype prefers a D(2h) paramagnetic triplet ground state due to Hund's rule, rather than a singlet. The highly polarized C-Be bonding weakens the rigidity of the tetrahedral carbon in T(d)C(BeH) 4 enormously, and the enhancement of both C-Be and Be 4 peripheral covalent bonding exerted by the extra electrons stabilizes the ptC eventually. The delocalization of the two p pi electrons is only modest, but their density on the most electronegative carbon atom helps stabilize the ptC arrangement. This is in contrast to the conventional strategy to delocalize p(pi) lone pairs for stabilizing the ptC arrangement. Various strategies to achieve neutral derivatives with ptCs are demonstrated.     

Improving the predictive quality of time-dependent density functional theory calculations of the X-ray emission spectroscopy of organic molecules

The simulation of X-ray emission spectra of organic molecules using time-dependent density functional theory (TDDFT) is explored. TDDFT calculations using standard hybrid exchange-correlation functionals in conjunction with large basis sets can predict accurate X-ray emission spectra provided an energy shift is applied to align the spectra with experiment. The relaxation of the orbitals in the intermediate state is an important factor, and neglect of this relaxation leads to considerably poorer predicted spectra. A short-range corrected functional is found to give emission energies that required a relatively small energy shift to align with experiment. However, increasing the amount of Hartree–Fock exchange in this functional to remove the need for any energy shift led to a deterioration in the quality of the calculated spectral profile. To predict accurate spectra without reference to experimental measurements, we use the CAM-B3LYP functional with the energy scale determined with reference to a Δself-consistent field calculation for the highest energy emission transition.     

Ab initio calculations of the lowest electronic states in the CuNO system

The lowest singlet and triplet electronic levels of the A' and A" symmetry species of the neutral copper-nitrosyl (CuNO) system are calculated by ab initio methods at the multi-reference configuration interaction (MRCI) level of theory with single and double excitations, and at the coupled cluster level of theory with both perturbational (CCSD(T)) and full inclusion of triple excitations (CCSDT). Experimental data are difficult to obtain, hence the importance of carrying out calculations as accurate as possible to address the structure and dynamics of this system. This paper aims at validating a theoretical protocol to develop global potential energy surfaces for transition metal nitrosyl complexes. For the MRCI calculations, the comparison of level energies at linear structures and their values from C(2v) and C(s) symmetry restricted calculations has allowed to obtain clear settings regarding atomic basis sizes, active orbital spaces and roots obtained at the multi-configurational self-consistent field (MCSCF) level of theory. It is shown that a complete active space involving 18 valence electrons, 11 molecular orbitals and the prior determination of 12 roots in the MCSCF calculation is needed for overall qualitatively correct results from the MRCI calculations. Atomic basis sets of the valence triple-zeta type are sufficient. The present calculations yield a bound singlet A' ground state for CuNO. The CCSD(T) calculations give a quantitatively more reliable account of electronic correlation close to equilibrium, while the MRCI energies allow to ensure the qualitative assessment needed for global potential energy surfaces. Relativistic coupled cluster calculations using the Douglas-Kroll-Hess Hamiltonian yield a dissociation energy of CuNO into Cu and NO to be (59 ± 5) kJ mol(-1) ((4940 ± 400) hc?cm(-1)). Favorable comparison is made with some of previous theoretical results and a few known experimental data.     

Correlation energy extrapolation by intrinsic scaling. I. Method and application to the neon atom

Remarkably accurate scaling relations are shown to exist between the correlation energy contributions from various excitation levels of the configuration interaction approach, considered as functions of the size of the correlating orbital space. These relationships are used to develop a method for extrapolating a sequence of smaller configuration interaction calculations to the full configuration-interaction energy. Calculations of the neon atom ground state with the Dunning's quadruple zeta basis set demonstrate the ability of the method to obtain benchmark quality results.     

Importance of spin-orbit effects on the isomerism profile of Au3: an ab initio study

Two-component relativistic density functional theory combined with high-level ab initio correlation techniques was applied to the study of the electronic structure and isomerism of Au(3). All calculations were performed with accurate small-core shape-consistent relativistic pseudopotentials. Density functional theory was used to determine the equilibrium structures of the Au(3) isomers and isomerization path and to estimate the contributions of spin-orbit effects to the ground state electronic energy along the path. The reliability of these estimates was verified through independent many-body multipartitioning perturbation theory calculations. Spin-orbit corrections were used to refine the isomerization energy profile computed by spin-orbit-free coupled cluster methods.     

The synergy between qualitative theory, quantitative calculations, and direct experiments in understanding, calculating, and measuring the energy differences between the lowest singlet and triplet states of organic diradicals

This perspective describes research, carried out in the authors' labs over the past forty years, aimed at understanding, predicting, and measuring the singlet-triplet energy differences (ΔE(ST)) in diradicals. A theory for qualitatively predicting the ground states of diradicals and the use of Negative Ion Photoelectron Spectroscopy (NIPES) for measuring ΔE(ST) are described. The application of this theory, ab initio calculations, and NIPES to the prediction and measurement of ΔE(ST) in a wide variety of organic diradicals is detailed. Among the diradicals that are discussed in this perspective are HN, CH(3)N, PhN, CH(2), trimethylenemethane (TMM), oxyallyl (OXA), meta-benzoquinodimethane (MBQDM), meta-benzoquinone (MBQ), tetramethyleneethane (TME), 1,2,4,5-tetramethylenebenzene (TMB), and D(8 h) cyclooctatetraene (COT). All of these diradicals have been studied in one and, in most cases, in both of the authors' laboratories. The studies of OXA and D(8h) COT were, in fact, collaborations between the research groups of the authors. These two projects both took advantage of the ability of NIPES to provide information about transition states. Transition-state spectroscopy was used to measure the carbonyl stretching frequency in the singlet state of OXA and to establish that D(8h) COT violates the strictest version of Hund's rule.     

Near threshold photoionization of the ground and first excited states of C(2)

Calculations using the multichannel Schwinger configuration-interaction method are presented for the photoionization from the ground and the first excited states of the C(2) molecule. Both single channel and multichannel calculations are presented in a photon energy range from the threshold to about 50 eV of photon energy. For the ground state, inclusion of both intrinsic and dynamical correlation effects is seen to strongly alter the picture of the photoionization process inferred from single-channel frozen-core Hartree-Fock calculations. Furthermore, the photoionization study of the first excited state of molecular carbon has revealed the presence of strong interchannel coupling between the 3sigma(g)-->ksigma(u) channel and the photoionization channels leading to the A (4)Pi(g) and f (2)Pi(g) ionic states in the near threshold region.     

Radial limit of lithium revisited

Configuration interaction (CI) calculations are carried out for the ground state of lithium using a thoroughly optimized basis set of s-type Slater functions. They establish that the radial limit of the nonrelativistic energy of the ground ²S state of lithium is no higher than −7.448666443E_h. Thus, radial correlation accounts for 35.2% of the total correlation energy. The radial CI wave function predicts a significantly more accurate Fermi contact parameter than the Hartree-Fock wave function. However, the imbalanced treatment of electron correlation in the radial CI wave function leads to an excessively diffuse electron density that is worse than that of the Hartree-Fock wave function. © 1997 John Wiley & Sons, Inc.