# Survey of Scientific Contributions, Klaus Ruedenberg

**2. Many Electron Theory and Electron Correlation**

**3. Analysis and Interpretation of Electronic Structure and Bonding **

**4. Analysis of Potential Energy Surfaces**

**5. Specific Chemical Reaction Surfaces**

Numbers in brackets refer to entries in the publication list

**Atomic orbital basis set**

** Even-tempered atomic orbitals, 1973-79** Introduction of practical primitive bases with systematic exponent sequences, avoiding premature linear dependencies, providing upper and lower bounds in atoms. Subsequently used for basis set constructions by Dunning, Huzinaga, Koga et al. [88, 95-100, 102, 103, 118, 119]

** General contractions, 1974-79** Introduced their use and developed algorithms avoiding the repetition of primitive integral evaluations. [102, 118, and Raffenetti, JCP 58, 4452 (1973)]

**Spherical harmonics**

*Translation of solid spherical harmonics and polypolar expansions, 1969***-**** 73** Used for orbital and multipole displacements. [60, 92, 176]

** Rotation of spherical harmonics, 1996**-

**Recursive construction of general rotation matrices, avoiding the instabilities and inefficiencies of Wigner's Euler-angles-based formulas. Used for rotations of orbitals and multipoles. Currently used worldwide outside quantum chemistry in a variety of computer imaging and animation software. [165, 175, 176]**

*2001***Orbital transformations and 2x2 Jacobi rotations**

*Resolution of general transformations into Jacobi rotation products, 1970***-**** 72** Resolution and reconstitution of non-singular N-dimensional matrices in terms of 2x2 rotations with a computational effort equal to that required for one matrix multiplication. Used for generating orbital transformations. [66, 82, 174]

** Induction of configurational transformations by orbital transformations, 1999** Generation of the transformations that are induced in N-electron configuration spaces by transformations of orbitals [174]

** Four-index transformations of electron repulsion integrals, 1993** Transformation of a set of electron-repulsions integrals by N(N-1)/2 sequential Jacobi rotations with a computational effort equal to that required for transformation by one NxN matrix. [152]

**Electron interaction energies between atomic orbitals**

** Exponential-type orbitals, 1951-72** This first charge-density-driven vectorization, achieved by certain formal transformations, reduced the scaling of the computational effort for a molecular integral set from the fourth to the second power of the orbital space dimension. It has remained the method of choice for high-accuracy work on linear molecules in terms of exponential-type orbitals. [4, 27, 55; other integral contributions: 5, 8, 40, 41, 44, 49, 50, 51, 52, 80, 81]

** Gaussian-type orbitals, 1970 **An integral package for Gaussian orbitals (Biggmoli by R.Raffenetti in the ALIS code).

**2. Many Electron Theory and Electron Correlation**

**Configuration interaction method**

** Symmetric-group-based configuration interaction with general configuration selection, 1971-74** A symmetric-group-based configuration interaction method and code, based on spin-adapted anti-symmetrized orbital products, allowing for arbitrary configuration selection. [75, 83, 85, 86, 87, 104]

** Determinant-based direct configuration interaction with general configuration selection, 2000-2001** A direct determinant-based configuration interaction method and fast code for large configuration spaces, applicable for full configuration spaces as well as for arbitrary configuration selections. Available in the GAMESS program system. [178]

** Dynamic correlation energy extrapolation through intrinsic scaling, 2003-ongoing** An extrapolation method (CEEIS) from double and triple excitations to full CI spaces that approximates the energies of large full configuration spaces within fractions of a millihartree by a computational effort reduced by many orders of magnitudes. The first practical full correlation energy recovery along reaction paths. [186 –189, 191, 198-200, 205]

**Multi-configuration self-consistent-field method**

** Self-consistent determination and correlation analysis of separated pair wave-functions, 1968-70 **The first self-consistent ab-initio separated pair wavefunctions. (For Li- to Ne+6 and LiH, BeH, NH.) Extensively used in Hurley's 1976 textbook

*Electron Correlation in Small Molecules.*[46-48, 63-65]

** General MCSCF approach based on super-CI expansion, 1976-81 **An early well-converging multi-configuration self-consistent field method and code based on orbital optimization by iterative improvements of natural orbitals through single-excitation annihilations. Popular with researchers experimenting with nonstandard CI expansions because of its capability to deal with arbitrary configuration selections. [109, 114, 116, 120]

** General MCSCF approach based on Jacobi-type orbital rotation sequences, 2003 **An MCSCF method for large arbitrary configuration spaces based on orbital optimizations of full rather than second order energy expressions. Available in the GAMESS program system. [181]

** Stable MCSCF method for excited states, 2003 **An MCSCF method for excited states in the presence of lower states of like symmetry, based on a functional that prevents root-flipping related instabilities. [181]

**Zeroth-order (non-dynamic) correlation and localized orbitals**

** Full optimized reaction space model, 1976-82 **Introduced the full optimized reaction space (FORS), generated by all reacting optimized valence orbitals, as an unbiased size-consistent configurational basis for zeroth-order reference functions in correlation calculations along reaction paths when major changes occur in the occupancies of dominating natural orbitals. Proved widely useful for elucidating chemical reactions and furnished a rigorous basis for ab-initio quantifications of atoms in molecules. [109, 114, 115, 117, 122, 137]

** A priori generation of full valence space wavefunctions free of configurational deadwood, 2000-ongoing **It is shown that only the determinants of a very small subspace contribute to the correlation recovered by the full configurational valence space when appropriate localized orbitals are used. A systematic method for the effective

*a-priori*determination of these correlating subspaces is developed, leading to a drastic shortening of the CI expansions in large molecules. [178, 179, 202, 203]

**Semi-empirical recovery of dynamic correlation**

** Intra-atomic correlation correction in the FORS model, 1985 **The FORS model, in its atom-localized form, proved perfectly suited for an ab-initio quantification of the Atoms-in-Molecules Correlation-Correction concepts, advanced by Moffit and Hurley's in the fifties and sixties to obtain an estimate of dynamic electron correlation contributions. [128, 129]

** A dynamic correlation energy estimate based on localized orbital, 2002 **A very simple formula for estimating dynamic correlation energies in terms of electron pair populations within and between the dominating localized orbitals yields correlation energies with a mean deviation of 2-3 kcal/mol for a large number of SCF dominated molecules. [180]

**3. Analysis and Interpretation of Electronic Structure and Bonding**

**Semi-empirical interpretations**

** Hückel model, topological model, tight-binding model and free-electron model, 1951-62** Demonstrated the rigorous isomorphism between these models, which are still useful for understanding large systems. Derived electronic spectra for cata- and peri-condensed hydrocarbon series. [6, 7, 9, 10, 11, 12, 13, 16-21, 24, 33]

** Total energy and orbital energies in SCF theory, 1977 **The SCF energy was shown to be approximately equal to 1.5 times the sum of the occupied orbital energies. Found particular resonance among density functional theorists. [110]

** Non-spherical atom densities in crystals from x-ray scattering data and chemical interpretation, 1985-97 **A method is devised for the deduction of chemical information from electron difference densities beyond the use of spherically averaged atomic densities. Applied to systems with high-accuracy diffraction data. [131, 138, 140, 141, 142, 143, 178]

**Physical understanding of covalent bond formation by electron sharing, 1962-1975, 2007-ongoing**

A physical interpretation of groundstate energy shifts is derived from the variation principle. The variational analysis shows covalent bond formation in H2+ to be the result of structural changes in the kinetic energy functional that are caused by the orbital delocalization due to electron sharing. They induce a firmer attachment of electrons to the nuclei, which accounts for the potential energy lowering identified by the virial theorem. Adopted in books by Mulliken, Fukui, Kutzelnigg and others. By 1986, one of the 100 most cited papers of the Reviews of Modern Physics. [22, 62, 69, 74, 108, 193, 194, 204]

**Correlation-adapted orbital localization**

*Intrinsic localized ab-initio self-consistent-field orbitals, 1963**-*** 77, 1993 **Three intrinsic localization criteria (energetic, harmonic, short-range) are specified and the definitive algorithmfor quantitative orbital localizations formulated. Quoted in Lipscomb's 1976 Nobel prize address as essential for his boron hydride work. By 1986 one of the 100 most cited papers of the Reviews of Modern Physics, a Current Contents Citation Classic. Applied to elucidate binding in a variety of molecules including delocalized pi-systems. [26, 37, 38, 42, 72, 73, 94, 101, 107, 111, 112, 152]

** Localized orbitals for strongly convergent configuration interaction expansions, 2003** Certain localized molecular orbitals are identified and generated that yield configuration interaction expansions converging more strongly than the previously believed optimal expansions generated from natural orbitals. [182]

**Atoms in molecules in rigorous ab-initio theory**

The model of molecules formed by bonded atoms, the functional concept derived from two hundred years of experimental chemistry, differs from the physical theory, which is based on electrons and nuclei. The derivation of an ab-initio quantification of the former from the latter, which would enhance the interaction between chemical experimentalists and quantum-chemical theorists, is still an on-going enterprise.

** Exact expansion of exact self-consistent-field orbitals in terms of quasi-atomic orbitals, 2003 **It is shown that the exact SCF (or DFT) density matrix can be exactly expanded in terms of a unique intrinsic minimal basis set of quasiatomic orbitals that deviate only slightly from free-atom orbitals. Also yields zeroth-order approximations to the most important correlation orbitals. Application to Bloch orbitals in crystals extracts quasi-atomic orbitals from plane-wave representations. [183, 184, 185, 190]

** Exact expansion of full valence-space MCSCF orbitals in terms of quasi-atomic orbitals, 1976-2003 **The full valence space MCSCF molecular orbitals are shown to be intrinsically expressible in terms of basis-set-independent molecular orbitals that have quasiatomic character. The many electron wavefunction is expressed in terms of quasi-atomic states. Quantitative measures for electron migration and charge transfer are obtained. [109, 122, 123, 124, 196, 197]

** Chemical orientation of quasiatomic orbitals, 2003 **An intrinsic criterion and algorithm is found that determines intra-atomic orthogonal transformations of the quasiatomic orbitals on all atoms in a molecule such that each of the resulting oriented quasiatomic hybrid orbitals is involved in at most one bond. [196, 197]

** Binding energy analysis in terms of quasi-atomic orbitals, 2007-ongoing** An intrinsic resolution of ab-initio binding energies in terms of interpretable interatomic contributions.

**4. Analysis of Potential Energy Surfaces**

**Reaction paths and reaction channels**

** Quadratic steepest descent, 1993-94 **An efficient true second-order algorithm is developed for integrating Euler's steepest descent differential equation, dispelling the unjustified stigma of the latter as a first-order method. Included [TCA 100, 21 (1998)] in the MOLPRO code. [156, 157, 158, 162]

** Gradient fields, gradient extremals and reaction channels, 1986-94 **Analyses of steepest descent fields show that intrinsic reaction paths, i.e. steepest descents from transition states to reactant and product minima, do not necessarily have the character of reaction channels. They do so only when running close to gradient extremals. The latter are new types of curves on potential energy surfaces whose properties are analyzed and for which a following-algorithm is developed. Also used for global transition state searches. [134, 153, 161]

** Critical Points, 1993-94 **The quadratic steepest descent method is shown to have advantages for minimum searches as well as transition state searches. Also, approximate transition states are predicted analytically using only the hessians from reactant and product minima. [158, 163, 164]

** Reaction path bifurcations, 1986 **A split into several branches complicates the identification of reaction paths. An analysis, achieved by introducing the valley-ridge inflection points, yields a classification of bifurcation patterns according to their location relative to transition states. [135]

**Intersections between potential energy surfaces of like symmetry**

** Violation of the weak non-crossing rule, 1990-1997 **According to the weak non-crossing rule, conical intersections between one-configuration-dominated closed-shell singlet states were considered extremely unlikely since they are impossible between single determinants, in particular when involving ground states. The analysis of ozone and its valence-isoelectronic analogues showed that such intersections do occur when the adiabatic states are superpositions of diabatic states that embody significant internal non-dynamic correlations varying markedly as the diabatic states switch dominance in the adiabatic states. [145, 150, 154, 171]

** Intersections of intersection seams, 1997-1999 **The analysis of the ozone case showed that intersection seams can consist of several branches that can cross at lower-dimensional nodes, so that the global intersection seam forms a topologically multi-connected network. This notably occurs when two crossing states having the same overall irreducible representation belong to different irreducible representations in a coordinate subspace of higher symmetry. [167, 173]

** Surface patterns near conical intersections, 1991 **The surface topographies in intersection regions are shown to be either of the sloped or the peaked type and are further classified by the characteristics of their singular gradient field patterns. Became a popular introduction to conical intersections. [151]

** Quantum chemical geometric phase theorem, 1990-1999 **A new derivation of the phase theorem without explicit reference to intersections. Entails a generalization to higher order intersections. Also developed was the use of the phase theorem for the ab-initio determination of conical intersections. [145, 150, 173]

** Diabatic States, 1993-1999 **A new ab-initio determination of diabatic states from adiabatic states based on enforcing configurational uniformity in the electronic wavefunction over the relevant region of the potential energy surface. [155, 170]

**Long-range interaction between atoms, 2007-ongoing**

It is shown how long-range interactions between bonded atoms can be resolved into multipole and dispersion interactions by means of resolving the configuration interaction wavefunction in terms of a zeroth-order component and a dynamic correlation component. [44, 201, 205]

**5. Specific Chemical Reaction Surfaces**

**Diatomic potential energy curves**

** Empirical regularities among spectroscopic constants of diatomic molecules, 1968** Accurate correlations are shown to exist between rotational and vibrational spectroscopic constants in 160 diatomic molecules. [54]

** Accurate ab-initio dissociation energies of C_{2}, N_{2}, O_{2}, F_{2}, 2005** Using the CEEIS method, extrapolating to a complete basis set and including relativistic effects, the dissociation energies of C

_{2}, N

_{2}, O

_{2}, F

_{2}are determined within 0.6 kcal/mol of the experimental values. [189]

** Accurate ab-initio potential energy curve of F_{2}, 2007-ongoing** Using the CEEIS method, extrapolating to a complete basis set and including relativistic effects, an accurate potential energy curve of F

_{2}is determined, which yields the entire vibrational spectrum with a mean absolute deviation of 5 cm

^{-1}. The long-range part is shown to be the superposition of quadrupole repulsions and dispersion attractions. [198, 199, 200, 205]

**Triatomic potential energy surfaces**

** Isomerization of nitrosyl-hydride, HNO NOH, 1977** Determination and atomic interpretation of the isomerization curves and barriers for the lowest singlet and triplet states by full-valence-space MCSCF calculations. [114 and Ph.D. thesis, M.G. Dombek, Iowa State Univ, 1977)]

*Global and local features of the lowest two singlet states of ozone, 1990-97 *** Isomerization O_{3} (obtuse angled) <—> O_{3} (equilateral ring). Conical intersection seam.** The ground state and the lowest excited state of the same symmetry are superpositions of the closed-shell s

^{14}p

^{4}and s

^{12}p

^{6}diabatic states, the former dominating at the open minima, the latter dominating at the ring-minimum of the ground state. The diabatic configurations switch dominance in the adiabatic wavefunctions near the transition states from the ring to the open structures of the ground state, and the excited state's global minima occur nearby. Within 0.04 Å of this region, the two states are degenerate, as proven by quantitative application of the phase theorem. This was the first instance of a conical intersection between two closed-shell singlets of like symmetry. The one-dimensional intersection seam was found to consist of four loops connected at three nodes. The global determination of both surfaces in the three-dimensional internal coordinate space shows them being close to each other over an extended region, containing the intersection seam, favoring radiationless transitions. Dissociation from the open ground state minimum into O

_{2}+O occurs by an end atom moving away with little change in the apex angle and a small barrier. [145, 150, 154, 166, 167, 168]

** Conical intersections in the molecules O_{3}, S_{3}, SO_{2}, S_{2}O, 1997 **All members of this valence-isoelectronic series are found to possess the same kind of conical intersection between the

^{1}A

_{1}ground state and the lowest excited

^{1}A

_{1}state as found in ozone. [171]

** Enthalpy of formation of water, 2006** The enthalpy of formation of water is recovered within 0.03 kc/mol. The energy of atomization is recovered within 0.06 kcal/mol. [191]

**Poly-atomic potential energy surfaces**

** Concerted dissociation of dioxetane, H_{2}COOCH_{2} 2 H_{2}CO, 1976** A full-valence-space calculation yielded the dissociation curve and an atomic interpretation of this Woodward-Hoffman-forbidden reaction. [109]

** Dissociation of ethylene H_{2}C=CH_{2} 2 CH_{2}, 1978-79** The first full-valence-space dissociation curve for ethylene, disproving the existence of a previously conjectured barrier for ethylene formation. [114, 117]

*Concerted dihydrogen exchange between ethylene and ethane, 1982
H _{3}C*

*-*

*CH*_{3}+ H_{2}C=CH_{2}

**Exhibits limits of the Woodward-Hoffmann rules by showing that this paradigm of a symmetry-allowed reaction has in fact a barrier of about 70 Kcal/mole. [125]**

*H*_{2}C=CH_{2}+ H_{3}C-CH_{3}*Ring opening of cyclopropylidene to allene, 1986**-*** 91
H_{2}(CCC)_{ring} H_{2} H_{2}C=C=CH_{2}, **The first global potential energy surface in terms of three primary internal coordinates, with twelve additional spectator coordinates optimized at each point in the primary coordinate space. The calculated 8 Kcal/mol barrier agrees with experimental inferences. The ring opening is not stereo-specific. The transition state maintains the reactant's C2v symmetry. The reaction path bifurcates after the transition state near an intersection with an excited state, yielding two stereo-isomeric lower symmetry products. On the downhill paths, the hydrogen pairs can perform free, but synchronized internal rotations. Analogous reactions of substituted reactants are stereo-specific due to non-bonded steric and long-range electrostatic interactions between the substituents. [136, 146, 147, 148, 149]

Robert S, Mulliken, Nobel Prize Winner, 1966 [43] Science, Art and Beauty, 1977 [113] Doctor Faustus, Person and Plays, 1993 [159] Robert S. Hansen, Director of the Ames Laboratory, 1999 [172]