Index-1 Saddle Points Research Articles

Global Optimization: Local Minima and Transition Points

We consider the minimization of smooth functions of the Euclidean space with a finite number of stationary points having moderate asymptotic behavior at infinity. The crucial role of transition points of first order (i.e., saddle points of index 1) is emphasized. It is shown that (generically) any two local minima can be connected via an alternating sequence of local minima and transition points of first order. In particular, the graph with local minima as its nodes and first order transition points representing the edges turns out to be connected (Theorem A). On the other hand, any connected (finite) graph can be realized in the above sense by means of a smooth function of three variables having a minimal number of stationary points (Theorem B).

Geometry, transition states, and vibrational spectra of boron isostere of N‐methylacetamide by ab initio calculations

AbstractThe basic unit of the peptide, the CONH bond, is responsible for imparting specific secondary structural features to peptides. Many modifications, such as isosteric/bioisosteric replacement of this basic unit, have been made to alter the properties of peptides. Isosteric replacement of the nitrogen atom by boron (boron peptides) is another plausible way of changing peptide characteristics. Like N‐methylacetamide (NMA), acetylmethylborane, or BMA, is a good model for studying boron peptides. The potential energy surface (PES) of BMA has been studied by Hartree–Fock (HF), density functional theory (DFT), and post‐HF methods. The PES of BMA is an inverted form of the NMA hypersurface. Two minima and two saddle points of index 1 have been identified on this PES and fully optimized at the HF, Becke's three‐parameter exchange functional and the gradient‐corrected functional of Lee, Yang, and Paar, second‐order Møller–Plesset (full), and quadratic configuration interaction method with single and double substitutions followed by a perturbative treatment of triple substitutions (QCISD) levels of theory. The minima correspond to structures with “omega” values close to 90° and 270°, and the transition states have omega values around 0° and 180°. The energy barrier for rotation around the “omega bond” is found to be 4.3 kcal/mol at the QCISD/6‐31G* level, which is much lower than the 16–23 kcal/mol barrier in NMA. The unique shape of the PES of BMA and the low barrier to rotation can be well explained by second‐order orbital interaction studies. Normal‐mode analysis reveals a significant shift in the BH stretching frequency of BMA from that in alkyl boranes and borane–phosphine complexes. The replacement of nitrogen by boron bestows on such peptides greater proteolytic stability, more flexibility around the omega angle, and a unique preference for positive ϕ angles, all of which are absent in peptides of natural origin. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

A Theoretical Study of the Molecular Mechanism for the Carboxylation Chemistry in Rubisco

Stereochemical and structural aspects for the carbon dioxide fixation followed by hydration of d-ribulose 1,5-bisphosphate catalyzed by Rubisco are analyzed by using two model substrates, hydroxypropanone and 3,4-dihydroxy-2-pentanone. The molecular mechanisms for the carboxylation, C3-hydration and C2-inversion processes, are theoretically characterized and discussed with saddle points of index 1 representing transition structures (TS) and relevant (molded) intermediates mirroring the experimentally proposed mechanism. Ab initio SCF MO calculations at 3-21G and 6-31G** basis set levels of theory were used, while the correlation energy is included at the MP2/6-31G** level. The mapping starts from TS1, the carboxylation transition structure, which describes now the coupling of the carbon dioxide attack to the substrate C2-center in its dienol form with a synchronous interconversion of the C3 hydroxyl into a ketone group. Thereafter, water addition leads to a gem-diol followed by another step of intramolecular hydrogen transfer coupled with the C2−C3 bond breaking process. This TS breaks into one model of 3-d-phosphoglycerate product and an intermediate. The configuration inversion at the C2-center is found to be possible via intramolecular hydrogen transfer, as suggested by the corresponding TS relating the intermediate to the C2-inverted conformation. The complete set of steps found gives a self-contained description of the carboxylation followed by hydrolysis with proper stereochemistry. Comparisons between the transition state analogue: 2-carboxy-d-arabinitol-1,5-bisphosphate and TS1 show that both structures can be superposed with minimal root-mean-square deviation for C-atoms. All other calculated stationary TSs share the gross conformational features of TS1, and consequently, all molecular rearrangements detected in a vacuum can be accommodated without constraints into the active site of Rubisco. Transition structures invariances with respect to level of theory and molecular models are discussed.

Transition-state structures for describing the enzyme-catalyzed mechanisms of rubisco

The carboxylation and oxygenation processes of a model substrate, 3,4-dihydroxy-2-pentanone, have been theoreticaly characterized as a set of steps, mimicking the corresponding reactions of D-ribulose-1,5-bisphosphate catalyzed by rubisco. A theoretical characterization is carried out of transition-state structures and possible molecular intermediates represented as saddle points of index 1 and minimum energy structures, respectively. The quantum chemical characterization, at the HF/3-21G calculation level, of these stationary points is used to rationalize and to discuss both catalyzed sequences. The reported set of these stationary points maps out most experimental aspects of the reaction pathways for the real system.

A simulated annealing based technique for locating first-order saddle points on multidimensional surfaces and constructing reaction paths: several model studies

We explore the workability of a simulated annealing based method for locating first-order saddle points (SP) on a potential energy surface, starting from a minimum. The search can proceed sequentially, exhausting all the saddle points of index one. It can also be tuned to trace the lowest first-order SP in one run. Four different model surfaces are investigated and the lowest first-order saddle point lying between two local minima are located in each case. Possible use of the technique in the construction of reaction paths is considered. A more realistic system involving six argon atoms interacting via Lennard-Jones potential (an 18 parameter problem) is investigated and the saddle point on the path of transformation of Ar 6 cluster from its global minimum configuration to a neighbouring local minimum is traced out. The computational labour in the SAM-based search is examined vis-a-vis the traditionally used methods.

Ab Initio Studies on N-Methylacetamide

The potential energy surface of N-methylacetamide (NMA) is investigated ab initio at the Hartree–Fock/6-31G* level with full geometry optimisation with or without MP2 correlation and thermal corrections in order to characterise its stationary points and the Intrinsic Reaction Coordinate (IRC) of trans-cis isomerisation. The cis and trans minima and the two non-equivalent Transition States (TS) of the isomerisation with opposite pyramidalisation of the amidic group are determined. The TS result considerably destablised (∼16 and 21 kcal mol-1) with respect to the most stable equilibrium structure. The lowest energy transition state is the up form (both substituents at N bent toward the carbonyl oxygen), while the down form (with the N lone pair pointing toward the carbonyl oxygen) is higher in energy. A saddle point of index 2 connects the two transition states, ∼8 kcal mol-1 above the TS up, the absence of intermediates being verified. Adding the correlation corrections does not change the energy gaps appreciably. The IRC at HF/6-31G* level has been followed in the most favourable case: correlate deformation of peptide unit and decoupling of methyl groups with respect to the reaction coordinate are shown. Additional TS, depending on methyl groups orientation, confirm the substantial free rotation around the phiv; and ψ torsional angles.

A quantum electronic theory of chemical processes?The inverted energy profile case: CH3+ + H2 reaction

A quantum approach to chemical processes is developed. The chemical interconversion is described as an electronic process. The reaction corresponds to histories involving quantum states belonging to different stationary molecular Hamiltonians. The system may be embedded in a weak (thermal) and/or external electromagnetic field. The electromagnetic transverse fields lead to transition moments yielding finite probability amplitudes for the system to change from one quantum state to another. Bottleneck subspaces (transition states) are defined; they mediate the interconversions in generic unimolecular and bimolecular processes. Active precursor and successor complexes are introduced to help bridge reactant and product electronic states. The stationary states are modeled with Born-Oppenheimer Hamiltonians. At a qualitative level, the theory is general. The rate, measured as a time derivative of product concentration, is expressed in terms of concentrations of active precursor and successor complexes. The kinetic coefficients are given in terms of quantum processes involving electronic bottleneck states. Stationary structures and vibrational zero-point energies characterizing the reactive CH3++H2 system are determined at a Hartree-Fock level of theory with 6-31++G** basis set. The vibrational levels are corrected with anharmonicity effects. The saddle point of index one for hydrogen scrambling reactions has been obtained and shown to be related to the CH5+ molecular complex together with the precursor and successor complexes geometries. The unusual properties of the system with respect to standard transition-state theory are fairly well described within this approach, in particular, isotope scrambling as well as photon emission during formation of the carbocation. The theory suggests that these types of reactions, which are found in outer space, may contribute to the scattering of the cosmic microwave background. © 1997 John Wiley & Sons, Inc.

A quantum electronic theory of chemical processes—The inverted energy profile case: CH3 + H2 reaction

A quantum approach to chemical processes is developed. The chemical interconversion is described as an electronic process. The reaction corresponds to histories involving quantum states belonging to different stationary molecular Hamiltonians. The system may be embedded in a weak (thermal) and/or external electromagnetic field. The electromagnetic transverse fields lead to transition moments yielding finite probability amplitudes for the system to change from one quantum state to another. Bottleneck subspaces (transition states) are defined; they mediate the interconversions in generic unimolecular and bimolecular processes. Active precursor and successor complexes are introduced to help bridge reactant and product electronic states. The stationary states are modeled with Born-Oppenheimer Hamiltonians. At a qualitative level, the theory is general. The rate, measured as a time derivative of product concentration, is expressed in terms of concentrations of active precursor and successor complexes. The kinetic coefficients are given in terms of quantum processes involving electronic bottleneck states. Stationary structures and vibrational zero-point energies characterizing the reactive CH3++H2 system are determined at a Hartree-Fock level of theory with 6-31++G** basis set. The vibrational levels are corrected with anharmonicity effects. The saddle point of index one for hydrogen scrambling reactions has been obtained and shown to be related to the CH5+ molecular complex together with the precursor and successor complexes geometries. The unusual properties of the system with respect to standard transition-state theory are fairly well described within this approach, in particular, isotope scrambling as well as photon emission during formation of the carbocation. The theory suggests that these types of reactions, which are found in outer space, may contribute to the scattering of the cosmic microwave background. © 1997 John Wiley & Sons, Inc.

Transition structures for hydride transfer reactions in vacuo and their role in enzyme catalysis

A general discussion as to the role of in vacuo transition structure in enzyme catalysis is presented. Quantum mechanical aspects are emphasized. The transition structures defined as saddle points of index one (SPi-1) for the hydride transfer step on different model enzyme systems from flavoproteins to dehydrogenases have been characterized with analytical gradients at different levels of theory: semi-empirical; ab initio with different basis sets within the Hartree-Fock scheme; density functional theory using different approaches. Quantum chemical characteristics of the SPi-1 are used to discuss hydride transfer step in enzyme catalyzed reactions and mechanistic implications. With the exception of dihydrofolate reductase, the results for all other systems studied suggest that the endo relative orientation imposed by the active site on the reactants is essential for polarizing the C dH t bond and situating the system in the quadratic region of the endo SPi-1. The geometry and transition vector components are both model independent and weakly dependent on the level of theory used in their determination. Comparisons of the SPi-1 geometries with available X-ray coordinates show that the SPi-1 can be fitted without any stress at the active site. The geometrical arrangement of the SPi-1 results in optimal frontier LUMO orbital interactions, and the transition vector amplitudes show primary and secondary isotope effects to be strongly coupled. A comparison between simple and sophisticated molecular models shows that there is a minimal molecular model associated with geometrical parameters describing the essentials of the chemical interconversion step. For hydride transfer, the corresponding transition vector is an invariant feature.

Transition structure and reactive complexes for hydride transfer in an isoalloxazine-nicotinamide complex. On the catalytic mechanism of glutathione reductase. An ab initio MO SCF study

Abstract An analysis is presented of the catalytic mechanism of glutathione reductase based upon a theoretically characterized saddle point of index one obtained for a model representing the active groups of the flavine and nicotinamide adenine dinucleotide phosphate, namely, an isoalloxazine and nicotinamide rings. The isoalloxazine rings appears deformed into a butterfly conformation in the saddle point of index one. The butterfly conformation is retained along the path leading to a reduced isoalloxazine (N5-H) forcing the transferred hydrogen to stick into the nicotinamide binding site, this geometric feature suggests the existence of a transposed hydride transfer path where the N5-proton goes back to a lysine residue leaving the electrons on FAD. This mechanism is discussed and proposed as an alternative catalytic pathway in glutathione reductase to the standard electron transfer one. These results help to understand the riddle created by the changing kinetic behavior of glutathione reductase when, for instance, 2,4,6-trinitro-benzene sulfonate is used to study in vitro kinetics or when specific site directed mutagenesis is performed.

Transition structures in vacuo and the theory of enzyme catalysis. Rubisco's catalytic mechanism: a paradigmatic case?

The molecular mechanism of Rubisco is discussed from the theoretical perspective provided by saddle points of index 1 (SPi-1) for the enolization, carboxylation and oxygenation of Rubisco's substrate, d-ribulose-1,5-bisphosphate, by using 3,4-dihydroxy-2-pentanone as substrate model and ab initio SCF MO at a 3-21G basis set level of theory. Intramolecular hydrogen transfer is shown to be a possible pathway for enolization. The geometric conformation of the in vacuo carboxylation transition structure is found to be superimposable on the transition state analogue bound to Rubisco. For oxygenation, the lowest energy SPi-1 has a triplet spin electronic structure and the carbon fragment has a large structural overlap with the C-fragment of the enolization and carboxylation SPi-1 structures. An explanation for the bifunctionality of the Rubisco enzyme arises from these structural similarities: once the catalytic process begins and the transition structure for the enolization step is reached, the system can catalyse both the carboxylation and the oxygenation processes with very little structural changes of the substrate moiety. Thus, depending on the gas molecule that enters into the active site, CO2 or O2, Rubisco will catalyse carboxilation or oxygenation of the substrate, respectively. The “inevitability” hypothesis for the oxygenation reaction is given theoretical support.

Transition structures in vacuo and the theory of enzyme catalysis. Rubisco's catalytic mechanism: a paradigmatic case?

The molecular mechanism of Rubisco is discussed from the theoretical perspective provided by saddle points of index 1 (SPi-1) for the enolization, carboxylation and oxygenation of Rubisco's substrate, d-ribulose-1,5-bisphosphate, by using 3,4-dihydroxy-2-pentanone as substrate model and ab initio SCF MO at a 3-21G basis set level of theory. Intramolecular hydrogen transfer is shown to be a possible pathway for enolization. The geometric conformation of the in vacuo carboxylation transition structure is found to be superimposable on the transition state analogue bound to Rubisco. For oxygenation, the lowest energy SPi-1 has a triplet spin electronic structure and the carbon fragment has a large structural overlap with the C-fragment of the enolization and carboxylation SPi-1 structures. An explanation for the bifunctionality of the Rubisco enzyme arises from these structural similarities: once the catalytic process begins and the transition structure for the enolization step is reached, the system can catalyse both the carboxylation and the oxygenation processes with very little structural changes of the substrate moiety. Thus, depending on the gas molecule that enters into the active site, CO 2 or O 2, Rubisco will catalyse carboxilation or oxygenation of the substrate, respectively. The “inevitability” hypothesis for the oxygenation reaction is given theoretical support.

On a quantum theory of chemical reactions and the role of in vacuum transition structures. Primary and secondary sources of enzyme catalysis

A general theoretical framework describing chemical reactions is presented. It is based on quantum mechanical concepts and a proper definition of an interconversion coordinate using elliptic coordinates. The quantum theory is integrated with general kinetic schemes. From the geometric and fluctuation vector properties derived from saddle point of index 1 a discussion of catalytic properties is given. A distinction can be made between primary and secondary sources of catalysis. The former are necessary and sufficient to produce a catalytic device. The latter are related to increasing (or decreasing) catalytic efficiency.

Potential nonrigidity of the NO3 radical

The equilibrium geometry and vertical excitation energies of the nitrate radical (NO3) have been determined at the coupled-cluster singles and doubles level. Unlike most previous theoretical methods used to study this problem, the present calculations circumvent the difficulties associated with symmetry-broken orbitals by using a reference function composed of orbitals obtained in a closed-shell self-consistent field (SCF) calculation for the NO3 anion. Although these orbitals do not satisfy the SCF equations for NO3 itself, they turn out to be more suitable for the correlation problem in the neutral molecule than the unrestricted Hartree–Fock solution for NO3. Nevertheless, our calculations agree with most previous studies in predicting that the high-symmetry D3h structure of NO3 is not a minimum on the potential energy surface. The potential near the D3h point is relatively flat and has seven stationary points: three (equivalent) C2v minima with one long and two short N–O bonds; three (equivalent) C2v transition states with two long and one short N–O bonds; and the D3h structure, which is unstable with respect to the in-plane degenerate mode (e′) and is consequently a saddle point of index two. Exchange of the oxygen positions via pseudorotation around the D3h stationary point is predicted to be an extremely facile process with a barrier height of ≊190 cm−1, suggesting that the molecule may be spectroscopically nonrigid, belonging to a molecule symmetry group which is isomorphic with D3h, as observed experimentally. Excitation energies are calculated for both the D3h structure and points on the pseudorotation pathway, in order to predict differences between values obtained from photodetachment spectroscopy of the NO3 anion and those determined by direct excitation of NO3.

Do supra-antara paths really exist for 2 + 2 cycloaddition reactions? Analytical computation of the MC-SCF Hessians for transition states of ethylene with ethylene, singlet oxygen, and ketene

The transition structures for the 2, + 2, cycloaddition of ethylene with ethylene, with singlet 02, and with ketene have been determined by MC-SCF gradient methods. The subsequent characterization of these critical points by analytical Hessian computation shows that in each case the critical point is a saddle point of index 2 (Le., there are two imaginary vibrational frequencies). Thus, the results show that the supra-antara reaction path may not exist at all for 2 + 2 cycloadditions.