# The CNDO equations

In a study of the effect of pressure on adsorption, for a system obeying the Langmuir equation (and Eq. XI-3, assuming ideal solutions and ai = 1), the value of K is 2.75 X l(y at 1 atm pressure and 1.23 x l(y at 3000 bar and 25°C. Consult the appropriate thermodynamic texts and calculate AV, the volume change for the adsorption process of Eq. XI-2. Comment on the physical significance of AV. [c.421]

When plotted according to the linear form of the BET equation, data for the adsorption of N2 on Graphon at 77 K give an intercept of 0.004 and a slope of 1.7 (both in cubic centimeters STP per gram). Calculate E assuming a molecular area of 16 for N2. Calculate also the heat of adsorption for the first layer (the heat of condensation of N2 is 1.3 kcal/mol). Would your answer for Vm be much different if the intercept were taken to be zero (and the slope the same) Comment briefly on the practical significance of your conclusion. [c.673]

The two-state equations of section B2.2.8.4 caimot, in general, be solved analytically except for the specific case of exact resonance when k. = kj= k and IJ.. = U, t/. = U. . Then the equations can be decoupled by introducing the linear combinations (R.) = J (R) Rj, so the two-state set can be converted to [c.2046]

In practice, however, the amount of hydrochloric acid employed is less than 5 per cent, of the amounts indicated by either of the above equations. Various explanations have been advanced to account for this one is that the following reaction is catalysed by acid or by hydroxonium ions [c.559]

Neglect of Differential Overlap (NDO) methods, such as CNDO and INDO, include electron repulsions, but the resulting equations are nonlinear. The energy calculation includes terms for pairs of electrons. These methods include nuclear repulsions, but only by reducing the charge on each nucleus by the number of core electrons shielding it. The other NDO methods, MINDO/3, MNDO, AMI, and PM3 replace nuclear repulsion terms in the potential energy by parameterized, core-repulsion terms. The terms compensate for considering only valence electrons in the electronic Schrodinger equation, and they incorporate effects of electron correlation (see Configuration Interaction on page 119). For more about these methods see Using Quantum Mechanics Methods on page 107 Using Quantum Mechanics Methods on page 107. [c.34]

Either of these approximations lead to a great simplification of the general equations, the greatest simplification being that associated with complete neglect of differential overlap (CNDO). An intermediate neglect of differential overlap (INDO, ZINDO/1, and ZINDO/S) scheme retains some of the one-center terms and the neglect of diatomic differential overlap (NDDO) schemes keep all of the one-center terms above. The modified INDO (MINDO/3) method and those that are based on NDDO (MNDO, AMI, and PM3) are all available in HyperChem and are discussed extensively below, in turn. First of all, however, we describe the results of SCF calculations that are independent of the details of whether the specific approximation involved is CNDO, MNDO, or some other approximate treatment of the general SCF equations. [c.239]

At this point a brief comment on the justification of testing the Gibbs or any other thermodynamically derived relationship is in order. First, it might be said that such activity is foolish because it amounts to an exhibition of scepticism of the validity of the laws of thermodynamics themselves, and surely they are no longer in doubt This is justifiable criticism in some specific instances but, in general, we feel it is not. The laws of thermodynamics are phenomenological laws about observable or operationally defined quantities, and where one of the more subtle deductions from these laws is involved it may not always be clear just what the operational definition of a given variable really is. This question comes up in connection with contact angles and the meaning of surface tensions of solid interfaces (see Section X-6). Second, thermodynamic derivations can involve the exercise of logic at a very rigorous level, and it is entirely possible for nqnsequiturs to creep in, which escape attention until an experimental disagreement forces a reexamination. Finally, the testing of a thermodynamic relationship may reveal unsuspected complexities in a system. Thus, referring to the preceding subsection, it took experiment to determine that the surface active species of Aerosol OTN was HX nither than (Na, X ) and that, Eq. III-93 was the appropriate form of the Gibbs equa-/fion to use. The difficulties in confirming the Kelvin equation for the case of liquids in capillaries have led people to consider various possible complexities (see Section III-1C). [c.79]

As with the BET equation, a number of modihcations of Eqs. XVII-77 or XVn-79 have been proposed, for example Ref. 71. FHH-type equations go to inhnite him thickness (i.e., bulk liquid), as P - F and this cannot be the case if the liquid does not wet the solid, and Adamson [72] proposed [c.628]

To convert (A1.1.37) into a quantum-mechanical fonn that describes the matter wave associated with a free particle travelling tlirough space, one might be tempted to simply make the substitutions v = Elh (Planck s hypothesis) and X = hip (de Broglie s hypothesis). It is relatively easy to verify that the resulting expression satisfies the time-dependent Sclirodinger equation. However, it should be emphasized that this is not a derivation, as there is no compelling reason to believe that this ad hoc procedure should yield one of the fiindamental equations of physics. Indeed, the time-dependent Sclirodinger equation caimot be derived in a rigorous way and therefore must be regarded as a postulate. [c.12]

At this point it is important to make some clarifying remarks (1) clearly one caimot regard dr in the above expression, strictly, as a mathematical differential. It caimot be infinitesimally small, since dr much be large enough to contain some particles of the gas. We suppose instead that dr is large enough to contain some particles of the gas but small compared with any important physical lengtii in the problem under consideration, such as a mean free path, or the length scale over which a physical quantity, such as a temperature, might vary. (2) The distribution fiinction / (r,v,t) typically does not describe the exact state of the gas in the sense that it tells us exactly how many particles are in the designated regions at the given time t. To obtain and use such an exact distribution fiinction one would need to follow the motion of the individual particles in the gas, that is, solve the mechanical equations for the system, and then do the proper countmg. Since this is clearly impossible for even a small number of particles in the container, we have to suppose that / is an ensemble average of the microscopic distribution fiinctions for a very large number of identically prepared systems. This, of course, implies that kinetic theory is a branch of the more general area of statistical mechanics. As a result of these two remarks, we should regard any distribution fiinction we use as an ensemble average rather than an exact expression for our particular system, and we should be carefiil when examining the variation of the distribution with space and time, to make sure that we are not too concerned with variations on spatial scales that are of the order or less than the size of a molecule, or on time scales that are of the order of the duration of a collision of a particle with a wall or of two or more particles with each other. [c.666]

Although the Sclirodinger equation associated witii the A + BC reactive collision has the same fonn as for the nonreactive scattering problem that we considered previously, it cannot he. solved by the coupled-channel expansion used then, as the reagent vibrational basis functions caimot directly describe the product region (for an expansion in a finite number of tenns). So instead we need to use alternative schemes of which there are many. [c.975]

This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. [c.1939]

The Fresnel equations predict that reflexion changes the polarization of light, measurement of which fonns the basis of ellipsometry [128]. Although more sensitive than SAR, it is not possible to solve the equations linking the measured parameters with n and d. in closed fonn, and hence they cannot be solved unambiguously, although their product yielding v (equation C2.14.48) appears to be robust. [c.2838]

Cases are known in which the use of the single substance surface tensions leads to predictions at variance with observation. For example, using equations (C2.14.49), (C2.14.50), (C2.14.51), (C2.14.52), (C2.14.53), (C2.14.54) and (C2.14.55) and the data in table C2.14.1 the interfacial free energy between seralbumin and silica is predicted to be positive, and the protein should therefore be repelled, whereas as is well known it is strongly adsorbed. There are several possible reasons for discrepancies. One is that the characteristic length scale of the (macroscopic) surface tension is different from (probably larger than) the characteristic length for protein adsorjDtion possibly it is more appropriate to use the microsurface tension [137] for these calculations. Another is that the use of average curvatures, surface potentials and so on is too cmde for biomolecules with their intricate surface topography. Finally, equation C2.14.49, equation C2.14.50, equation C2.14.51, equation C2.14.52, equation C2.14.53, equation C2.14.54 and equation C2.14.55 assume that the protein remains unchanged upon interaction with the surface. Wlrile this is likely to be tme up to the moment of initial contact, native folded proteins are only marginally stable and intramolecular contacts maintaining the native stmcture may be substituted by molecule-surface contacts with [c.2840]

Finally, there is the case that the elechonic set i j.(r,R) is not a complete set. Then, neither fI (r,R) in Eq. (90), nor the nuclear equation (91) is exact. Moreover, the truncated Lagrangean in Eq. (96) is not exact either and this shows up by its not possessing a full symmetry (viz., lacking invariance under local gauge transformation). We can (and should) remedy this by introducing a YM field, which is not now a pure gauge field. This means that the internally induced YM field cannot be transformed away by a (local) gauge transformation and that it brings in (through the back door, so to speak) the effect of the excluded electronic states on the nuclear states, these being now dynamically coupled between themselves. [c.152]

The merging of NM analysis and MD is attractive because of the complementarity of the two techniques. The former provides an accurate description for an equilibrium reference system, while the latter can yield in theory a complete description of nonequilibrium events. Such a union has been mainly pursued in the context of essential dynamics [81, 82] in which major features of macromolecules are approximately described by following a small set of low-frequency vibrational modes. The problem in this description arises from the need to identify this low-frequency subset from dynamic simulations (via correlation matrices) a dominant set of low-frequency modes cannot, in general, be reliably estimated from the short simulations feasible today [83], given the long relaxation times of proteins in solution. Therefore, techniques based on projecting the low-frequency motion on the Newton equations of motion [81, 82], or splitting harmonic and anharmonic motion [84] with limited harmonic-model updating, cannot work for general biomolecular applications. Deviations from the harmonic approximation can emerge within 15 fs [73]. For the same reasons, a realistic description of a protein s long-time dynamics based on a low-frequency vibrational subset might only be possible through substantial incorporation of information from enhanced sampling techniques [82]. [c.246]

Unfortunately, this local error Cr cannot be calculated, since we do not know the exact solution to the QCMD equations. The clue to this problem is given by the introduction of an approximation to Let us consider another discrete evolution with an order q > p and define an error estimation via er t + z i) - z t). [c.403]

It is true that the structure, energy, and many properties ofa molecule can be described by the Schrodingcr equation. However, this equation quite often cannot be solved in a straightforward manner, or its solution would require large amounts of computation time that are at present beyond reach, This is even more true for chemical reactions. Only the simplest reactions can be calculated in a rigorous manner, others require a scries of approximations, and most arc still beyond an exact quantum mechanical treatment, particularly as concerns the influence of reaction conditions such as solvent, temperature, or catalyst. [c.2]

They represent the generalization of our earlier equations (11.12) and include an additional equation for the pressure, since this can no longer be assumed to take the constant value p throughout the pellet. The differential equation relating X to f is derived exactly as before and has the same form, except that p cannot be replaced by p In the present case. Thus we have [c.118]

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form [c.55]

The summation runs over the N, objects in cluster i, each located at q and where the mean of the cluster is r,-. The total information loss is calculated by adding together the values for each cluster. At each iteration that pair of clusters which gives rise to the smallest increase in the total error function are merged. Two more hierarchical clustering algorithms are the centtoid method, which determines the distance between two clusters as the distance between their centroids, and the median method, which represents each cluster by the coordinates of the median value. Fortunately, all six hierarchical agglomerative methods can be represented by a single equation, first proposed by Lance and Williams [Lance and Williams 1967], with the different algorithms having different coefficients. [c.511]

Apart from the prediction of a variable viscosity, generalized Newtonian constitutive models cannot explain other phenomena such as recoil, stress relaxation, stress overshoot and extrudate swell which are commonly observed in polymer processing flows. These effects have a significant impact on the product quality in polymer processing and they should not be ignored. Theoretically, all of these phenomena can be considered as the result of the material having a combination of the properties of elastic solids and viscous fluids. Therefore mathematical modelling of polymer processing flows should, ideally, be based on the use of viscoelastic constitutive equations. Formulation of the constitutive equations for viscoelastic fluids has been the subject of a considerable amount of research over many decades. Details of the derivation of the viscoelastic constitutive equations and their classification are covered in many textbooks and review papers (see Tanner, 1985 Bird et al, 1977 Mitsoulis, 1990). Despite these efforts and the proliferation of proposed viscoelastic constitutive equations in recent years, the problem of selecting one which can yield verifiable results for a fluid under all types of flow condition.s is still unresolved (Pearson, 1994). In practice, therefore, the remaining option is to choose a constitutive viscoela.stic model that can predict the most dominant features of the fluid behaviour for a given flow situation. It should also be mentioned here that the use of a computationally costly and complex viscoelastic model in situations that are different from those assumed in the formulation of that model will in general yield unreliable predictions and should be avoided. [c.9]

As described in Chapter 1, mathematical models that represent polymer flow systems are, in general, based on non-linear partial differential equations and cannot be solved by analytical techniques. Therefore, in general, these equations are solved using numerical methods. Numerical solutions of the differential equations arising in engineering problems are usually based on finite difference, finite element, boundary element or finite volume schemes. Other numerical techniques such as the spectral expansions or newly emerged mesh independent methods may also be used to solve governing equations of specific types of engineering problems. Numerous examples of the successful application of these methods in the computer modelling of realistic field problems can be found in the literature. All of these methods have strengths and weaknesses and a number of factors should be considered before deciding in favour of the application of a particular method to the modelling of a process. The most important factors in this respect, are type of the governing equations of the process, geometry of the process domain, nature of the boundary conditions, required accuracy of the calculations and computational cost. [c.17]

Simplification achieved by using a constant mesh in the modelling of the flow field in a single-blade mixer is not applicable to twin-blade mixers. Although the model equations in both simulations are identical the solution algorithm for twin-blade mixers cannot be based on the VOF method on a fixed domain and instead the Arbitrary Lagrangian-Eulerian (ALE) approach, described in Chapter 3, Section 5.2, should be used. However, the overall geometry of the plane of the rotors blades cross-section is known and all of the required mesh configurations can be generated in advance and stored in a file to speed up the calculations. Figure 5.4 shows the finite element mesh corresponding to 19 successive time steps from the start of the simulation in a typical twin-blade tangential rotor mixer. The finite element mesh configurations correspond to counter-rotating blades with unequal rotational velocities set to generate an mieven stress field for enhancing dispersive mixing efficiency. Calculation of mesh velocity, required for modification of the free surface equation (see Equation (3.73)) at each time step, is based on the following equations (Ghoreishy, 1997) [c.146]

Obviously selection of the most efficient elements in conjunction with the most appropriate finite element scheme is of the outmost importance in any given analysis. However, satisfaction of the criteria set by these considerations cannot guarantee or even determine the overall accuracy, cost and general efficiency of the finite element simulations, which depend more than any other factor on the algorithm used to solve the global equations. To achieve a high level of accuracy in the simulation of realistic problems usually a refined mesh consisting of hrmdrcds or even thousands of elements is used. In comparison to time spent on the solution of the global set, the time required for evaluation and assembly of elemental stiffness equations is small. Therefore, as the number of equations in the global set grows larger by mesh refinement the computational time (and hence cost) becomes more and more dependent on the effectiveness and speed of the solver routine. The development of fast and accurate computational procedures for the solution of algebraic sets of equations has been an active area of research for many decades and a number of very efficient algorithms are now available. [c.199]

Computer simulation of non-isothermal, non-Newtonian flow processes starts with the formulation of a mathematical model consisting of the governing equations, arising from the laws of conservation of mass, energy, momentum and rheology which describe the constitutive behaviour of the fluid, together with a set of appropriate boundary conditions. The formulated mathematical model is then solved via a computer based numerical technique. Therefore, the development of computer models for non-Newtonian flow regimes in polymer processing is a multi-disciplinary task in which numerical analysis, computer programming, Quid mechanics and rheology each form an important part. It is evident that these subjects cannot be covered in a single text, and an in-depth description of each area requires separate volumes. It is not, however, realistic to assume that before embarking on a project in the area of computer modelling of polymer processing one should acquire a thorough theoretical knowledge in all of these subjects. Indeed the normal time period allowed for completion of [c.287]

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. [c.169]

We cannot solve the Schroedinger equation in closed fomi for most systems. We have exact solutions for the energy E and the wave function (1/ for only a few of the simplest systems. In the general case, we must accept approximate solutions. The picture is not bleak, however, because approximate solutions are getting systematically better under the impact of contemporary advances in computer hardware and software. We may anticipate an exciting future in this fast-paced field. [c.169]

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. [c.172]

The reason the Schroedinger equation for molecules cannot be separated appears in the last term, involving a sum of repulsive energies between electrons. To [c.175]

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. [c.10]

Statistical mechanics is the mathematical means to calculate the thermodynamic properties of bulk materials from a molecular description of the materials. Much of statistical mechanics is still at the paper-and-pencil stage of theory. Since quantum mechanicians cannot exactly solve the Schrodinger equation yet, statistical mechanicians do not really have even a starting point for a truly rigorous treatment. In spite of this limitation, some very useful results for bulk materials can be obtained. [c.12]

Some reactions, such as ion-molecule association reactions, have no energy barrier. These reactions cannot be described well by the Arrhenius equation or [c.164]

Nitric acid being the solvent, terms involving its concentration cannot enter the rate equation. This form of the rate equation is consistent with reaction via molecular nitric acid, or any species whose concentration throughout the reaction bears a constant ratio to the stoichiometric concentration of nitric acid. In the latter case the nitrating agent may account for any fraction of the total concentration of acid, provided that it is formed quickly relative to the speed of nitration. More detailed information about the mechanism was obtained from the effects of certain added species on the rate of reaction. [c.8]

To direct the synthesis so that only Phe Gly is formed the ammo group of phe nylalanme and the carboxyl group of glycine must be protected so that they cannot react under the conditions of peptide bond formation We can represent the peptide bond for matron step by the following equation where X and Y are amine and carboxyl protecting groups respectively [c.1136]

The CNDO/INDO, MINDO/3, Z3NDO/1, and ZINDO/S methods might be expected to imply an even simpler equation for the electron density than the above. For example, a rigorous complete neglect of CNDO approximation, suggests that equations (87) and (88) should be replaced by expressions with a sum only over diagonal elements of the density matrix. This would represent a molecular charge density that is the exact sum of atomic densities. Alter- [c.242]

See pages that mention the term

**The CNDO equations**:

**[c.273] [c.136] [c.668] [c.32] [c.40] [c.390] [c.687] [c.242] [c.273] [c.120] [c.722] [c.27] [c.49] [c.243]**

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