# Complete set

Solution First, we must construct the balanced composite curves using the complete set of data from Table 7.1. Figure 7.5 shows the balanced composite curves. Note that the steam has been incorporated within the construction of the hot composite curve to maintain the monotonic nature of composite curves. The same is true of the cooling water in the cold composite curve. Figure 7.5 also shows the curves divided into enthalpy intervals where there is either a [c.220]

The Champ-Sons model is a most effieient tool allowing quantitative predictions of the field radiated by arbitrary transducers and possibly complex interfaces. It allows one to easily define the complete set of transducer characteristics (shape of the piezoelectric element, planar or focused lens, contact or immersion, single or multi-element), the excitation pulse (possibly an experimentally measured signal), to define the characteristics of the testing configuration (geometry of the piece, transducer position relatively to the piece, characteristics of both the coupling medium and the piece), and finally to define the calculation to run (field-points position, acoustical quantity considered). [c.737]

Note that equation (A3.11.1881 includes a quantum mechanical trace, which implies a sum over states. The states used for this evaluation are arbitrary as long as they form a complete set and many choices have been considered in recent work. Much of this work has been based on wavepackets [46] or grid point basis frmctions [47]. [c.993]

With the frequency removed from the sum, (B1.1.9) has just a sum over vibrational integrals. Because all the vibrational wavefiinctions for a given potential surface will fomi a complete set, it is possible to apply a sum rule to simplify the resulting expression [c.1130]

Given the complete set of solutions to this one-electron equation, a complete set of A -electron mean-field wavefiinctions can be written. Each 4 J. )is constructed by fomiing a product of orbitals chosen from the set [c.2162]

Let us denote a state of the system by y. For the purposes of discussion, we shall concentrate on a system composed of atoms, and for this y represents the complete set of coordinates r =(rp r2,.. . r ) and conjugate [c.2243]

Selecting trial moves in an unbiased way typically means (a) choose an atom randomly , with equal probability from the complete set (b) displace it by random amounts in the v, y and z directions, chosen [c.2257]

Both the electronic and the geometry optimization problem, particularly the latter, may have more than one solution. For small, rigid molecules, tire approximate molecular geometry is chemically obvious, and the presence of multiple minima is not a serious concern. For large, flexible molecules, however, finding the absolute minimum, or a complete set of low-lying equilibrium structures, is only a partially solved problem. This topic will be discussed in the last section of this chapter. The rest of the article deals with local optimization, i.e., finding a minimum from a reasonably close starting point. We will also discuss the detennination of other stationary points—most importantly saddle points-constrained optimization, and reaction paths. Several reviews have been published on geometry optimization [1, 2]. The optimization of SCF-type wavefiinctions is often highly nonlinear, particularly for the multiconfigurational case, and this has received most attention [3, 4]. [c.2332]

If the intermediate summations are over a complete set, then [c.158]

As written, Eq. (52) depends on all the (infinite number of) adiabatic electi onic states. Fortunately, the inverse dependence of the coupling strength on energy separation means that it is possible to separate the complete set of states into manifolds that effeetively do not interact with one another. In particular, Baer has recendy shown [54] that Eq. (57), and hence Eq. (58) also holds in the subset of mutually coupled states. This finding has important consequences for the use of diabatic states explored below. [c.278]

As the eigenfunctions form a complete set [c.314]

While this derivation uses a complete set of adiabatic states, it has been shown [54] that this equation is also valid in a subset of mutually coupled states that do not interact with the other states. [c.314]

In the following, it shall always be assumed that the zeroth-order solution is known, that is, we have a complete set of eigenvalues and wave functions, labeled by the electronic quantum number n, which satisfy [c.403]

A complete set of trihalides for arsenic, antimony and bismuth can be prepared by the direct combination of the elements although other methods of preparation can sometimes be used. The vigour of the direct combination reaction for a given metal decreases from fluorine to iodine (except in the case of bismuth which does not react readily with fluorine) and for a given halogen, from arsenic to bismuth. [c.213]

Ch em uses Slater atom ic orbitals to con struct sent i-em pirical molecular orbitals. I he complete set of Slater atomic orbitals is called the basis set. Core orbitals are assumed to be chemically inactive and arc not treated explicitly. Core orbitals and the atomic nucleus form the atomic core. [c.43]

Let us now assemble the complete set of dimensionless parameters for the problem. These are set out in Table 11.1, where the last column indicates the nature of their dependence on the external pressure p, the mean pore diameter and the pellet radius a. Symbols ft and 0 [c.125]

The true value of tk for a many-electron atom or a molecule is unknown. If we could set it equal ( expand it) to a linear combination of an infinite number of basis functions, each defined in a space of infinite dimensions, we could carry out an exact calculation of (k. Such a set of basis functions would be a complete set. [c.242]

The various basis sets used in a calculation of the H and S integrals for a system are attempts to obtain a basis set that is as close as possible to a complete set but to stay within practical limits set by the speed and memory of contemporary computers. One immediately notices that the enterprise is directly dependent on the capabilities of available computers, which have become more powerful over the past several decades. The size and complexity of basis sets in common use have increased accordingly. Whatever basis set we choose, however, we are attempting to strike a balance. If the basis set is too small, it is inaeeurate if it is too large, it exceeds the capabilities of our computer. Whether our basis set is large or small, if we attempt to calculate all the H and S integrals in the secular matrix without any infusion of empirical information, the procedure is described as ab initio. [c.242]

It should be mentioned that if two operators do not commute, they may still have some eigenfunctions in common, but they will not have a complete set of simultaneous eigenfunctions. For example, the and Lx components of the angular momentum operator do not commute however, a wavefunction with L=0 (i.e., an S-state) is an eigenfunction of both operators. [c.47]

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function

*[c.57]*

Perturbation theory is the second most widely used approximation method in quantum chemistry. It allows one to estimate the splittings and shifts in energy levels and changes in wavefunctions that occur when an external field (e.g., an electric or magnetic field or a field that is due to a surrounding set of ligands - a crystal field) or a field arising when a previously-ignored term in the Hamiltonian is applied to a species whose unperturbed states are known. These perturbations in energies and wavefunctions are expressed in terms of the (complete) set of unperturbed eigenstates. [c.59]

The algebraic methods of reconstruction give result at incomplete and complete set of initial projection data. But the iterative imhlementation of these methods requires large computing resources. Algebraic method can be used in cases, when the required accuracy is not great. [c.219]

In any experiment, tlie probability of observing a particular non-degenerate value for the system property A can be detemiined by the following procedure. First, expand the wavefiinction in temis of the complete set of nomialized eigenfimctions of the qiiantmn-niechanical operator, [c.9]

The choice of basis fiinctions is straightforward in atomic calculations. It can be demonstrated that all solutions to an independent-particle Hamiltonian have the synnnetry properties of the hydrogenic wavefiinctions. Each is, or can be written as, an eigenfiinction of die and i operators and involves a radial part multiplied by a spherical hamionic. Atomic calculations that use basis sets (not all of them do) typically choose fiiiictioiis that are similar to those that solve the Sclirodinger equation for hydrogen. If the complete set of hydrogenic fiiiictioiis is used, the sohidon to the basis set equations are the exact Hartree-Fock solutions. However, practical considerations require the use of finite basis sets the corresponding solutions are therefore only approximate. Although the distinction is rarely made, it is preferable to refer to these as self-consistent field (SCF) solutions and energies in order to distinguish them from the exact Hartree-Fock results. As the quality of a basis is improved, the energy approaches diat of the Hartree-Fock solution from above. [c.33]

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the [c.140]

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. [c.140]

Because of (equation Al.4.107) and because of the fact that / y aiid f z conumite with each other, we know that there exists a complete set of simultaneous eigenfimctions of Py-, Py, / y and ft. An eigenfiinction oi Py, P Y and P has the fonn [c.166]

The Bom-Oppenlieimer theory is soundly based in that it can be derived from a Schrodinger equation describing the kinetic energies of all electrons and of all N nuclei plus the Coulomb potential energies of interaction among all electrons and nuclei. By expanding tire wavefiinction that is an eigenfiinction of this fidl Sclirodinger equation in the complete set of fiinctions and then neglecting all tenns that involve derivatives of any j with respect to the nuclear positions Q ], one can separate variables such tliat [c.2155]

E.( ), possesses a complete set of eigenfiinctions, the matrix F whose dimension Mis equal to the number of atomic basis orbitals, has M eigenvalues e. and M eigenvectors whose elements are the. . Thus, there are [c.2170]

To completely specify the orientational ordering, the complete set of orientational order parameters, P/,L = 0,2,4.. ., is required. Only the even rank order parameters are non-zero for phases with a symmetry plane perjDendicular to the director (e.g. N and SmA phases). [c.2555]

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]

We have ignored a term m (0c0fc — 0j,0c) u), which is zero by the commutativity of derivatives. The crucial step is now, as in [72] and in other later derivations, the evaluation of the fifth and sixth terms by insertion of k) i (which is the unity operator, when k is summed over a complete set) [c.154]

[c.1128]

See pages that mention the term

Computational chemistry using the PC (2003) -- [ c.242 ]

Perturbation theory is the second most widely used approximation method in quantum chemistry. It allows one to estimate the splittings and shifts in energy levels and changes in wavefunctions that occur when an external field (e.g., an electric or magnetic field or a field that is due to a surrounding set of ligands - a crystal field) or a field arising when a previously-ignored term in the Hamiltonian is applied to a species whose unperturbed states are known. These perturbations in energies and wavefunctions are expressed in terms of the (complete) set of unperturbed eigenstates. [c.59]

The algebraic methods of reconstruction give result at incomplete and complete set of initial projection data. But the iterative imhlementation of these methods requires large computing resources. Algebraic method can be used in cases, when the required accuracy is not great. [c.219]

In any experiment, tlie probability of observing a particular non-degenerate value for the system property A can be detemiined by the following procedure. First, expand the wavefiinction in temis of the complete set of nomialized eigenfimctions of the qiiantmn-niechanical operator, [c.9]

The choice of basis fiinctions is straightforward in atomic calculations. It can be demonstrated that all solutions to an independent-particle Hamiltonian have the synnnetry properties of the hydrogenic wavefiinctions. Each is, or can be written as, an eigenfiinction of die and i operators and involves a radial part multiplied by a spherical hamionic. Atomic calculations that use basis sets (not all of them do) typically choose fiiiictioiis that are similar to those that solve the Sclirodinger equation for hydrogen. If the complete set of hydrogenic fiiiictioiis is used, the sohidon to the basis set equations are the exact Hartree-Fock solutions. However, practical considerations require the use of finite basis sets the corresponding solutions are therefore only approximate. Although the distinction is rarely made, it is preferable to refer to these as self-consistent field (SCF) solutions and energies in order to distinguish them from the exact Hartree-Fock results. As the quality of a basis is improved, the energy approaches diat of the Hartree-Fock solution from above. [c.33]

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the [c.140]

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. [c.140]

Because of (equation Al.4.107) and because of the fact that / y aiid f z conumite with each other, we know that there exists a complete set of simultaneous eigenfimctions of Py-, Py, / y and ft. An eigenfiinction oi Py, P Y and P has the fonn [c.166]

The Bom-Oppenlieimer theory is soundly based in that it can be derived from a Schrodinger equation describing the kinetic energies of all electrons and of all N nuclei plus the Coulomb potential energies of interaction among all electrons and nuclei. By expanding tire wavefiinction that is an eigenfiinction of this fidl Sclirodinger equation in the complete set of fiinctions and then neglecting all tenns that involve derivatives of any j with respect to the nuclear positions Q ], one can separate variables such tliat [c.2155]

E.( ), possesses a complete set of eigenfiinctions, the matrix F whose dimension Mis equal to the number of atomic basis orbitals, has M eigenvalues e. and M eigenvectors whose elements are the. . Thus, there are [c.2170]

To completely specify the orientational ordering, the complete set of orientational order parameters, P/,L = 0,2,4.. ., is required. Only the even rank order parameters are non-zero for phases with a symmetry plane perjDendicular to the director (e.g. N and SmA phases). [c.2555]

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]

We have ignored a term m (0c0fc — 0j,0c) u), which is zero by the commutativity of derivatives. The crucial step is now, as in [72] and in other later derivations, the evaluation of the fifth and sixth terms by insertion of k) i (which is the unity operator, when k is summed over a complete set) [c.154]

[c.1128]

See pages that mention the term

**Complete set**:**[c.13] [c.40] [c.42] [c.160] [c.218] [c.454] [c.1314] [c.1320] [c.2162] [c.152] [c.194] [c.557] [c.147] [c.44] [c.47] [c.64]**Computational chemistry using the PC (2003) -- [ c.242 ]