Ground-state multiplicity by spin-extended Hartree–Fock method

New possibilities of the spin-extended Hartree–Fock method in determining the sequence of energy levels are analyzed and demonstrated by the example of some oxygen-contained compounds.     

Study of potential curves by UHF-type methods. VI. Spin-extended Hartree–Fock method

An improved computational procedure of the spin-extended Hartree–Fock method is presented. The main features of the EHF potential curves for the dissociation of the LiH, BeH, and Li₂ molecules are discussed. The results are compared with curves calculated by the perturbation or CI methods using the same basis set.     

Some developments in the spin-extended Hartree-Fock method

All the second-order density matrix spin components for the spin-extended Hartree-Fock method are obtained. The coefficients in the final formulae are only ω_sM, ω_sM±1, ω_sM±2, where ω_sM are the weights of pure states of spin s in the initial unprojected determinant with spin projection M. The eigenvalue problem for the best electron density natural orbitals in the spin-extended method is formulated. All the second-order transition density matrix spin components between pure spin basis functions built of orthogonal orbitals and distinguished by different core choice are also found. This basis may be used on CI calculations.     

Matrix formulation of the generalized Hartree-Fock methods

Spin-projected one-particle density and spin density matrices are presented as polynomials of suitable unprojected quantities with generalized Sasaki-Ohno coefficients. Thus an explicit form of Harriman's theorems is given. For the two-particle spatial density matrix an expansion in direct products of powers of unprojected residual electron and spin density matrices is given. For these basic matrices of the scheme the variational spin-extended equations are formulated.     

Projected electron density method of π-eletron systems II. Excited states

A perturbation theory based on the time-dependent Schrödinger equation is presented; Coulombic interactions are taken into account and spin properties are neglected. Using wave functions given by the projected electron density method described in Part I as a basis set the energies of excited π-electron states are calculated. For a series of porphyrin compounds the electronic spectra are calculated and are found to be in good agreement with experiment.     

The spin-polarized extended Brueckner orbitals

Conventional natural and Brueckner orbitals (BOs) are rather frequently used for improving active orbital spaces in various configuration interaction (CI) approaches. However, the natural and Brueckner single-determinant models per se fail to give an adequate picture of highly correlated and quasidegenerate states such as open-shell singlet and dissociative states. We suggest the use of the spin-polarized extended BOs formally defining them in the same manner as in Lo?wdin's spin-extended Hartree-Fock method. Such BO orbitals turn out to be quite flexible and particularly useful for analyzing highly correlated electronic states. It is shown that the extended BOs always exist, unlike the usual unrestricted BOs. We discuss difficulties related to violation of size-consistency for spin projected determinant models. The working algorithm is proposed for computing BOs within the full CI and related complete active space methodology. The extended BOs are analyzed in terms of the special density-like matrices associated with spin-up and spin-down BO orbitals. From these density matrices, the corresponding spin-polarization diagrams are produced for effectively unpaired (essentially correlated) electrons. We illustrate the approach by calculations on cyclic hydrogen clusters (H(4), H(6), and H(8)), certain carbene diradicals and monoradicals, and low-lying excited states. The computations show that the BO spin-projected determinant provides a strong overlap with the multi-configurational state even for quasidegenerate states and bond breaking processes.     

On the efficiency of different EHF wavefunctions

A new computational scheme for the spin-extended Hartree-Fock (EHF) equations is shown to be suitable for large Π-electron systems. It is demostrated that the triplet EHF wavefunction with equal numbers of orbitals of different spins gives a much larger energy decrease than that with the maximum total-spin projection value.     

The generalized valence bond π orbitals of ethylene and allyl cation

Using a fixed sigma core obtained from full electron ab initio Hartree-Fock calculations, the spatially projected GVB orbitals for the pi electron systems of ethylene and allyl cation are reported. The GVB(SP) method generates wavefunctions possessing the correct spatial and spin symmetry without restricting the nature of the individual orbitals. The GVB(SP) wavefunction provides a simple interpretation of the molecule in terms of orbitals each containing a single electron. The resulting total energies and excitation energies agree very well with full configuration interaction calculations.     

Communication: ROHF theory made simple

Restricted open-shell Hartree-Fock (ROHF) theory is formulated as a projected self-consistent unrestricted HF (UHF) model by mathematically constraining spin density eigenvalues. This constrained UHF (CUHF) wave function is identical to that obtained from Roothaan's effective Fock operator. The α and β CUHF Fock operators are parameter-free and have eigenvalues (orbital energies) that are physically meaningful as in UHF, except for eliminating spin contamination. This new way of solving ROHF leads to orbitals that turn out to be identical to semicanonical orbitals. The present approach removes ambiguities in ROHF orbital energies.     

Configurational correlations in the coverage dependent adsorption energies of oxygen atoms on late transition metal fcc(111) surfaces

The coverage dependence of oxygen adsorption energies on the fcc(111) surfaces of seven different transition metals (Rh, Ir, Pd, Pt, Cu, Au, and Ag) is demonstrated through density functional theory calculations on 20 configurations ranging from one to five adsorption sites and coverages up to 1 ML. Atom projected densities of states are used to demonstrate that the d-band mediated adsorption mechanism is responsible for the coverage dependence of the adsorption energies. This common bonding mechanism results in a linear correlation that relates the adsorption energies of each adsorbate configuration across different metal surfaces to each other. The slope of this correlation is shown to be related to the characteristics of the valence d-orbitals and band structure of the surface metal atoms. Additionally, it is shown that geometric similarity of the configurations is essential to observe the configurational correlations.     

Broken orbital symmetry and the description of valence hole states in the tetrahedral [CrO4]2− anion

The localization of ligand-based valence holes in the tetrahedral complex ion [CrO₄]^2? in a crystalline environment is studied by SCF calculations on the hole states, with progressively less restrictions on the spatial symmetry of the molecular orbitals. The final wavefunctions are obtained by constructing, from the symmetry broken SCF solutions, wavefunctions that exhibit again the proper transformation properties under the operations of T _d . The crystal environment of the [CrO₄]^2? anion is represented by a point charge model. In contrast with the situation for core hole states, the projection afterwards into T _d symmetry is important. The final ionization energies, which are obtained from projected C _3v adapted SCF solutions, are reduced considerably (?3 eV) with respect to the T _d ΔSCF results, but the ordering of the states has not changed essentially. The calculated ionization energies compare favourably with results of XPS experiments on Na₂CrO₄. The evaluation of the energies of projected symmetry broken SCF solutions requires the calculation of hamiltonian matrix elements between determinantal wavefunctions built from mutually non-orthogonal orbital sets. An efficient method for the calculation of such matrix elements is presented.     

An efficient self-consistent field method for large systems of weakly interacting components

An efficient method for removing the self-consistent field (SCF) diagonalization bottleneck is proposed for systems of weakly interacting components. The method is based on the equations of the locally projected SCF for molecular interactions (SCF MI) which utilize absolutely localized nonorthogonal molecular orbitals expanded in local subsets of the atomic basis set. A generalization of direct inversion in the iterative subspace for nonorthogonal molecular orbitals is formulated to increase the rate of convergence of the SCF MI equations. Single Roothaan step perturbative corrections are developed to improve the accuracy of the SCF MI energies. The resulting energies closely reproduce the conventional SCF energy. Extensive test calculations are performed on water clusters up to several hundred molecules. Compared to conventional SCF, speedups of the order of (N/O)2 have been achieved for the diagonalization step, where N is the size of the atomic orbital basis, and O is the number of occupied molecular orbitals.     

Quantum mechanical investigation of the inner-sphere reorganization energy of cyclooctatetraene/cyclooctatetraene radical anion. Part I

The inner-sphere reorganization energy of the electron self-exchange of the couple cyclooctatetraene/cyclooctatetraene radical anion has been investigated by quantum mechanical calculations. The more stable Jahn Teller distorted B2g conformation of the radical anion has been used in this study. Two different theories have been applied in this first part. The harmonic approximation in the classical Marcus scheme has been modified by using projected force constants, which are obtained from the complete force constant matrix and the geometry changes of the molecule during the ET (introduced by Mikkelsen). A different approach (introduced by Nelsen) combines the different energies of the neutral and radical anion with and without relaxation corresponding to the vertical ionization potential and the vertical electron affinity. The electronic energies of the neutral molecule and the radical anion differ dramatically applying three different levels of quantum mechanical calculations (UAM1, UB3LYP, PMP2 with three different basis sets with and without diffuse functions). Nevertheless the Nelsen method gives almost consistent results for the inner-sphere reorganization energies: 120.1 kJ/mol for semiempirical UAM1 method, 159.3 kJ/mol, 156.4 kJ/mol and 158.3 kJ/mol for density functional UB3LYP/6-31G*, UB3LYP/6-31++G* and UB3LYP/AUG-cc-pVDZ calculations and 192.5 kJ/mol for ab-initio PMP2/6-31G* investigations, respectively. These values are in agreement with earlier experimental work supposing the total reorganization energy to be larger than 38 kcal/mol assuming an electron self-exchange rate of 10(4) M(-1) s(-1). The simple harmonic approximation of Marcus relation has not yet been applied for a molecule like cyclooctatetraene with large torsional geometry changes. Using the projected force constants after scaling, considerably different results for the inner-sphere reorganization energy have been calculated: 738.1 kJ/mol for the UB3LYP/6-31G*, 743.3 kJ/mol for UB3LYP/6-31++G* and 759.1 kJ/mol for UB3LYP/AUG-cc-pVDZ level of theory. Comparison with our concentration dependent EPR experiments are controversial to the earlier experimental results, but the latter supports the assumption that the electron self-exchange occurs in a time scale so that the molecules cannot complete their vibrational motions. Therefore the projected Marcus relation is not valid for cyclooctatetraene/cyclooctatetraene radical anion including a large torsional change during the electron transfer.     

Thermisch stimulierte Lumineszenzerscheinungen. I Einfluß der Vorbehandlung von Kupferstaub und das Phänomen „Kontaktgedächtnis”︁

Thermally Stimulated Luminescence Phenomena. I. Influence of Pretreatment of Copper Dust and the Phenomenon “Contact Memory” The thermally stimulated luminescence of so-called “copper dust” is strongly influenced by its pretreatment. Samples which have been cooled prior to measurement show lower energies and those which have been mechanically treated, irradiated or heated, show higher thermoluminescence energies than samples which have been stored normally. After removal of the samples from the metallic carrier and subsequent heating of the latter, the thermoluminescence energies are proportional to those found for the sample measured immediately before. This phenomenon is considered as “contact memory”. All results are interpreted according to the concept of the hierarchic order.     

Dispersion energy evaluated by using locally projected occupied and excited molecular orbitals for molecular interaction

The dispersion terms are evaluated with the perturbation theory based on the locally projected molecular orbitals. A series of model systems, including some of the S22 set, is examined, and the calculated binding energies are compared with the published results. The basis set dependence is also examined. The dispersion energy correction is evaluated by taking into account the double excitations only of the dispersion type electron configurations and is added to the 3rd order single excitation perturbation energy, which is a good approximation to the counterpoise (CP) corrected Hartree-Fock (HF) binding energy. The procedure is the approximate "CP corrected HF + D" method. It ensures that the evaluated binding energy is approximately free of the basis set superposition error without the CP procedure. If the augmented basis functions are used, the evaluated binding energies for the predominantly dispersion-bound systems, such as rare gas dimers and halogen bonded clusters, agree with those of the reference calculations within 1 kcal mol(-1) (4 kJ mol(-1)). The limitation of the present method is also discussed.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

Tensor factorizations of local second-order Møller-Plesset theory

Efficient electronic structure methods can be built around efficient tensor representations of the wavefunction. Here we first describe a general view of tensor factorization for the compact representation of electronic wavefunctions. Next, we use this language to construct a low-complexity representation of the doubles amplitudes in local second-order M?ller-Plesset perturbation theory. We introduce two approximations--the direct orbital-specific virtual approximation and the full orbital-specific virtual approximation. In these approximations, each occupied orbital is associated with a small set of correlating virtual orbitals. Conceptually, the representation lies between the projected atomic orbital representation in Pulay-Saeb? local correlation theories and pair natural orbital correlation theories. We have tested the orbital-specific virtual approximations on a variety of systems and properties including total energies, reaction energies, and potential energy curves. Compared to the Pulay-Saeb? ansatz, we find that these approximations exhibit favorable accuracy and computational times while yielding smooth potential energy curves.     

苯胺和乙二醇生成吲哚反应机理的DFT理论研究

利用传统过渡态理论研究了苯胺和乙二醇生成吲哚反应的机理,设计了四种可能的反应通道.优化计算了反应物、产物和每个通道中可能的中间体和过渡态的几何构型,找出并确认了每个反应通道中各基元反应的可能过渡态和活化能.经分析对比给出该反应可能的主反应通道.同时初步探讨了在经过渡态TS1的基元反应中加入催化剂Ag后的反应机理.加入催化剂后,反应的活化能降低.     

Scanning‐Tunneling‐Spectroscopy‐Directed Design of Tailored Deep‐Blue Emitters

Frontier molecular orbitals can be visualized and selectively set to achieve blue phosphorescent metal complexes. For this purpose, the HOMOs and LUMOs of tridentate Pt^II complexes were measured using scanning tunneling microscopy and spectroscopy. The introduction of electron‐accepting or ‐donating moieties enables independent tuning of the frontier orbital energies, and the measured HOMO–LUMO gaps are reproduced by DFT calculations. The energy gaps correlate with the measured and the calculated energies of the emissive triplet states and the experimental luminescence wavelengths. This synergetic interplay between synthesis, microscopy, and spectroscopy enabled the design and realization of a deep‐blue triplet emitter. Finding and tuning the electronic “set screws” at molecular level constitutes a useful experimental method towards an in‐depth understanding and rational design of optoelectronic materials with tailored excited state energies and defined frontier‐orbital properties.