Double primed

D(co) = 7t(co)/ao J(c ) = maximum amplitude all have storage (single prime) and loss (double prime) components... 

For the dynamic components the storage component is designated by a single prime, and the loss component by a double prime. 

The use of single ( ) and double (") primes to indicate the upper and lower states, respectively, of a transition is general in spectroscopy and will apply throughout the book. 

For this reason we want to determine the number of centric and bicentric trees separately. Numbers of centric trees will be distinguished from those of bicentric trees by primes and double primes, respectively. Of the p, a, T, respectively, free trees are... 

The evolution of the probability distribution over time consists of adiabatic development and stochastic transitions due to perturbations from the reservoir. As above, use a single prime to denote the adiabatic development in time A, r —> r, and a double prime to denote the final stochastic position due to the influence of the reservoir, T —> T . The conditional stochastic transition probability may be taken to be... 

The double prime superscript is used to emphasize the basis of unit surface area. 

The starting point for London s treatment is a partitioning of the N electrons and v nuclei of a composite Hamiltonian H into two subunits (distinguished by prime and double-prime symbols) associated with respective Hamiltonian operators H and H",... 

The double-primed summation is over all distinguishable arrangements of a set of vertices of fixed composition, n, in a cycle, like vertices being treated as indistinguishable. 

The double prime on the summation over <5(n) signifies that it is over every arrangement of vertices in each of the 9-bond chains when only vertices corresponding to different defect species are distinguishable. The summation over n gives terms corresponding to every possible composition of each of the v chains. To carry out the summations over n and <5(n) we again employ the properties of the Fourier transforms and of the matrix S2. 

In deriving Eqs. (454) and (455), we have explicitly used the factorization property of pf>aB for large separations the double prime in Eq. (455) indicates an irreducibility condition similar to the one used in Eq. (430), except that the words on its left have to be replaced by "on its right in the definition after Eq. (432). 

Here, single prime refers to the feed side (high pressure) and double prime refers to the product side (low pressure) and the lower case c s refer to the bulk concentrations. Even if the membrane is homogeneous, it is possible to have a variable diffu-sivlty, which may be a function of salt concentration. 

In the text, the force-mass-length-time system of units has been used, with exceptions noted. Primed symbols refer to actual average values, as distinct from superficial or apparent values. A double prime indicates a point value. 

Accdg tp Ref 2, fugacity is a quantity which measures the true escaping tendency of a gas, a sort of idealized pressure. If primes and double primes refer, respectively, id an ideal gas and a real gas, then dF V dp = RTdlnp, and dF" = v"dp = RTdlnf,... 

J 90° and 180° both produce equivalent configurations of the molecule. A rotation through 270° also produces an equivalent configuration, but that is equivalent to a rotation through 90° in the opposite direction and so does not indicate an extra type of axis of symmetry. The molecular plane contains two C2 axes (the superscript prime is to distinguish them from the C2 axis which is coincident with the C4 axis), both of which contain the iodine atom and two diametrically opposed chlorine atoms. The two C2 axes are contained respectively by the two vertical planes which also contain the major axis. There are, in addition, two C2 axes (coincident with the x and y axes) which are contained by the horizontal plane, are perpendicular to the major axis, and also bisect the respective Cl-I-Cl angles. The double-prime superscript serves to distinguish these axes from the C2 and C2 axes. The C2 axes are contained by two extra planes of symmetry (they are the xz and yz planes), which are termed dihedral planes since they bisect the two C2 axes. The dihedral planes contain the major axis, C4. 

Note that in Figs. 2 and 3 the prime indicates that converged BS orbitals were provided as initial guess, whereas the double prime denotes that converged high-spin orbitals were employed. The energy of the closed-shell restricted state denoted as S=R has been arbitrarily set to zero. 

If the point group has a oh operation but no i operation (groups nb and B with n odd) the labels are primed or double primed according to whether the character of oh is positive or negative, respectively. The situation is similar to the one in the previous paragraph in that oh will be represented by +1 or -1 times an identity matrix and that there are twice as many irreducible representations in B as in [Pg.132]

Ue in (6.7), the value at the equilibrium geometry of the potential energy for nuclear motion. The convention is to denote quantities of the upper electronic state of a transition by single primes, and quantities of the lower electronic state by double primes. The symbols Te, G, and F are used for the electronic, vibrational, and rotational energies in cm-1 ... 

Primes and double primes are attached to all letters, when appropriate, to indicate those which are, respectively, symmetric and antisymmetric with respect to ah. 

In Table I we give the irreducible representations, in Mulliken s notation, contained in the central and attached orbitals for compounds of other types of symmetry. When 3 or 4 atoms are trigonally or tetragonally attached, we have supposed that the plane of these atoms is a plane of symmetry, as in (N03) or Ni(CN)4 When there is no such symmetry plane, as in NHg, the distinctions between u and g, or between primes and double primes, are to be aholished,9 and the symmetries degenerate to Csv, Civ instead of DSh, D h- When 6 atoms are attached in the scheme Z>3, or 8 in % they are arranged respectively at the corners of a trigonal and a square prism. 

In rare cases in which a third ring compound is attached to one of the intermediate N atoms of an open chain nitrogen compound, its substituents would be numbered clockwise using double primes, as 1",2 ",3 , etc. In some cases the groups NH2, N02, etc may be attached to intermediate N atoms of the open N chain. Following is the formula of a complicated hypothetical compound ... 

---