Computational details

In the following sections, the computational details of the theoretical calculations performed in this study, followed by the results derived from those, will be described. 

All the theoretical calculations in this study were carried out at the DFT level by means of the dispersion-corrected M06 functional [33] and using the Gaussian09 program [34]. We selected this functional in purpose because it has been shown to deliver good results for both organometallic systems and non-covalent interactions [33]. The mechanistic study was performed on the very Pd catalyst used for experiments, thus without simplifying the phosphine ligands bound to the metal center. Moreover, for the ZnMeCl reactant two additional explicit solvent molecules (i.e. THF) were included to fulfill the coordination sphere of Zn. The choice of this 

Solvent effects (i.e. THF, s = 7.4257) were introduced by a discrete-continuum model two THF molecules were explicitly included in the calculations as potential ligands (see above), and the effect of the bulk solvent was considered with a continuum method, the PCM approach [39], by means of single point calculations at all optimized gas phase geometries. In this method, the radii of the spheres employed to create the cavity for the solute were defined with the UFF model,. which is the default in Gaussian09. 

Unless otherwise specified, all the energies shown in this study correspond to Gibbs energies in THF (AGthf) at 223 K, obtained by employing the following scheme  

All conventional coupled-cluster calculations were performed with the Mainz-Austin-Budapest version of the Aces II program [83] and with the MRCC program [89, 90]. Within all post Hartree-Fock calculations, both core and valence electrons were correlated (all-electron calculations). For the open-shell cases, the UHF reference wave function was used. The Hartree-Fock and GGSD(T) contributions to the IPs and EAs were computed within the correlation-consistent doubly augmented quintuple-d-aug-cc-pwCV5Z, basis 

All explicitly correlated calculations were performed at the CCSD(F12) level of theory, as implemented in the TurbomOLE program [58, 69]. The Slater-type correlation factor was used with the exponent 7 = 1.0 aQ. It was approximated by a linear combination of six Gaussian functions with linear and nonlinear coefficients taken from Ref. [44]. The CCSD(F12) electronic energies were computed in an all-electron calculation with the d-aug-cc-pwCV5Z basis set [97]. For all cases we used full CCSD(F12) model (see Subsection 4.9 for the discussion about models implemented in Turbomole), the open-shell species were computed with a UHF reference wave function. The explicitly correlated contributions to the relative quantities are collected in Tables 10 and 11 under the label F12 . 

The mass-velocity and Darwin terms (denoted as MVD ) were computed at the ae-CCSD(T) level of theory [145] with the d-aug-cc-pwCV5Z basis set. The spin-orbit corrections ( SO ) were obtained from the experimentally observed spin-orbit splitting [159]. The diagonal Born-Oppenheimer correction was was crudely estimated from the total 

The aug-cc-pwCV5Z density-fitting [93] and aug-cc-pV5Z exchange-fitting [94] basis sets [the latter one used for the representation of the MP2-F12 intermediate, see 

The atomic ionization potentials and electron affinities are presented in Tables 10 and 11, respectively. In the first eight columns the incremental contributions to the final quantities are given, in the column labeled with Calc. the sum of them is shown. The last column, denoted with Exptl. , contains the experimental results. All values are provided in eV units. In the case of the nitrogen anion, an excess electron is unbound, therefore no EA is presented for this case. 

In all cases the calculations were performed using QM/MM methodology that includes the pseudobond model for the QM/MM boundary [13,39,41]. This methodology has been implemented in a modified version of Gaussian 98 [42], which interfaces to a modified version of TINKER [43], The AMBER94 all-atom force field parameter set [44] and the TIP3P [45] model for water were used. 

As explained above, we compare four different reaction paths for the first step of the reaction catalyzed by 40T. The first difference between the paths was the method employed to determine the MEP for the paths. The first two paths (A and B), were determined by the combined procedure [27, 35], In both cases seven images were employed for the optimization of the path. Path C was calculated with the CD method. Finally, path D corresponds to the MEP obtained from our previous calculations [33], 

The second major difference between the paths was the initial enzyme environment employed for the path determinations. The environment for path A corresponds to a previously optimized MEP [25], where the initial enzyme environment was 

In order to perform FEP calculations on paths A and B, the modified NEB method presented in Section 3.2.3 was employed. In both cases two extra images were added between the optimized images. This resulted in a total of 19 images for both paths. 

FEP calculations for paths A, B and C were performed with a 40 ps equilibration run prior to the sampling for all points along the path. The free energy contributions were sampled for 20 ps for each point on the MEP. In all cases a time step of 2.0 fs was employed, maintaining a constant temperature of 300 K. The SHAKE [47] algorithm was used to constrain all bonds involving hydrogen atoms. 

The reference energies and geometries used in Table 1.2 were computed with counterpoise-corrected CCSD(T /CBS//MP2/ 

The geometrical errors shown in Table 1.3 include 6 N-H- 0 bond lengths and angles and 11 N-H N bond lengths and angles. fix(-H) Y and y denote the bond length between X-Y and H-Y, 

2 Basis set, molecular orbitals, configuration state functions 

The electronic wavepackets are determined in an expansion with the configuration state functions (CSF) of single and double excitations (CISD) with the STO-6G basis set. This basis set is obviously small, but the main concern in this work is not the accm-acy but qualitative insights about the dynamical electrons in the course of chemical reactions. The program codes for the semiclassical Ehrenfest theory (SET) have been implemented in the GAMESS package [357]. 

The initial geometry for formamide was set to the enol form as in Fig. 7.10. The atoms O, C, and N make a molecular plane, and the bridging water molecule is also placed initially so as to lie in that plane. In what follows, we refer to the molecular orbitals approximately lying on the plane and to those approximately perpendicular to the plane as a and tt orbitals, respectively. Likewise, using only tt orbitals in 7(r,f) and Bab (r.f)) we estimate the tt electron density and tt bond-order, respectively. Similarly, the a electron density and a bond-order are made available. This distinction between the a and tt subspaces is just a matter of convenience, and of course they are not physical observables individually, since Cs symmetry is not imposed on the molecular system. On the contrary, all the vibrational modes are active in the present SET calculations. Since the aim of this study is not to estimate the reaction probability but the mechanism of the electron dynamics associated with proton transfer, we chose somewhat artificial initial conditions of nuclear motion to sample as many paths achieving proton transfer as possible. 

Minima and transition states (TSs) were optimized using the state specific complete active-space self-consistent field (SS-CASSCF) method. MECIs were optimized using state-averaged CASSCF (SA-CASSCF) [71] with equal weighting given to the two states forming the Cl. 

The orbitals included in these active spaces are represented in Fig. 3.1. Hereafter, the lira /Kis State will be simply labelled lira.  

For the MEO optimizations using the CAS 1 active space, we found it necessary to use a quadratically convergent algorithm for the CASSCF wavefunction. The orbital rotation derivative contributions from the coupled perturbed multi-configurational self-consistent field (CP-MCSCF) equations (see Refs. [82-84] and the Chap. 3 of Ref. [85] for details) were neglected in the MECI optimizations using the CAS2 active space. 

The notation SAn-CASSCF is used to describe a state averaged CASSCE calculation using orbitals averaged over n electronic states. 

D uj) is the Fourier transform of the dipole moment D t) which is calculated at every step from the electron density [13]  

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Including triply excited configurations is often needed in order to obtain very accurate results with MP4, QCISD or CCSD (see Appendix A for some of the computational details). The following example illustrates this effect. 

In principle, the Knn could be deduced from electronic structure calculations, but the smallest ones amount only to fractions of meV, whereas the calculations deal with binding energies, i.e. some eV it can be understood why the calculation techniques are not yet sufficiently accurate to compute detailed interactions, and why we find it better, until now, to extract them from local order parameters ... 

The treatment in this chapter has been theoretical. For a brief, dear, and very practical description of computational details for a number of standard problems, [10] is unsurpassed, and [12] can be recommended for programming techniques for automatic computers. For information on ordinary differential equations, the reader should consult [2], and for partial differential equations, [1]. For general methods of reduction to algebraic form as well as methods of solution, see [5], [7], and [8]. 

Studied in another publication (Koulouris and Stephanopoulos, 1995). Some computational details and variations of the adaptation algorithm have also to be evaluated to improve the efficiency of the computations. 

The calculations of the and c constants lead to a system of linear equations similar to that of the SCF-CI method, but with three more lines and columns corresponding to the coupling of the polynomial function with the electric field perturbation. The methodology and computational details have already been discussed (1) we stress two points the role of the dipolar factor, the nature and the number of the exeited states to inelude in the summation. 

Finally, an important advantage of the Gauss-Newton method is also that at the end of the estimation besides the best parameter estimates their covariance matrix is also readily available without any additional computations. Details will be given in Chapter 11. 

The organization of the paper is as follows. In the Methods Section we present a brief review of the procedures employed to determine the MEP, namely QM/MM geometry optimizations and the CD method (Section 3.2.1). NEB and the second order parallel path optimizer methods are reviewed in Section 3.2.2. In Section 3.2.3 we present a modified NEB method which allows the addition of extra images to a converged path. Finally we describe the FEP method developed in our lab in Section 3.2.4. In the Computational Details Section we present a description of the procedures employed for the determination of all four paths. Subsequently, we analyze the results obtained from all four paths and conclude with closing remarks. 

This article gives a simple introduction to the electron densities of molecules and how they can be analyzed to obtain information on bonding and geometry. More detailed discussions can be found in the books by Bader (4), Popelier (5), and Gillespie and Popelier (6). Computational details to reproduce the results presented in this paper are presented in Appendix 1. 

Having suggested the connections between relaxation descriptors and the data it is now important to realize that here is sufficient information in isochronal scans that, with numerical analysis now readily carried out by computer, detailed parameters that describe relaxation can be determined jointly. Analysis is most conveniently carried out with the aid of a parameterized empirical phenomenological function. The method as implemented by us uses for each relaxation peak a Cole- Cole -like function (4) to represent the complex modulus,... 

This set of articles presents the computational details and actual values for each of the statistical methods shown for collaborative tests. These methods include the use of precision and estimated accuracy comparisons, ANOVA tests, Student s t-testing, The Rank Test for Method Comparison, and the Efficient Comparison of Methods tests. From using these statistical tests the following conclusions can be derived ... 

Figure 21. The (So — S2) absorption spectrum of pyrazine for the reduced three-dimensional model using different spawning thresholds. Full line Exact quantum mechanical results. Dashed line Multiple spawning results for — 2.5, 5.0, 10, and 20. (All other computational details are as in Fig. 20.) As the spawning threshold is increased, the number of spawned basis functions decreases, the numerical effort decreases, and the accuracy of the result deteriorates (slowly). In this case, the final size of the basis set (at t — 0.5 ps) varies from 860 for 0 = 2.5 to 285 for 0 = 20.

A Positive/negative sign indicates increased decreased relative barriers and reaction energies, respectively, relative to the generic [Ni0(r 2-butadiene)2PII3] catalyst. For computational details see Reference lib. 

In the following, we give a brief description on how to evaluate the various terms in Equation 1. A more extensive summary of the computational details typically used in our studies as well as the CPU requirements of the MM/PBSA approach can be found elsewhere.11 simply evaluates the average potential energy of the system using the same force field (e.g. Equation 3 of Cornell et al.)12 as used to propagate the molecular dynamics trajectory. 

The computational details for both the quantum mechanical and FEP studies using molecular dynamics simulations are described elsewhere.9 In brief, the gas-phase free energies (AGgas) were calculated using energies... 

Many different procedures have been published, all of them aimed at finding the characteristic values of the parameters m, ]i and A = R2(E — l/i )/2, needed to produce acceptable solutions to the coupled equations. With the allowed values of m known, the procedure consists in finding the relation that must exist between A and ]i to produce an acceptable solution of the r] equation, and using this relation to calculate from the equation characteristic values of A and hence of the energy. The computational details are less important and have often been reduced to reliable computer routines that yield the precise results[85], best represented in terms of binding energy curves, such as those shown below for the ground and first excited states. 

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