Bipolar expansion

The electrostatic, induction, and dispersion terms can be expanded in a convergent series closely related to the multipole expansion, but fully accounting for the charge-overlap effects, the so-called bipolar expansion introduced by Buehler and Hirschfelder199,200. In the local coordinate systems with the origins located at the centers of masses of the monomers A and B, separated by the distance R, and with their x and y axes parallel and aligned along the z axes, the distance between two particles in space can be expressed as follows,... 

The expansion given above is exact, except for r12 = 0. If the terms resulting from the regions II-IV are neglected, one recovers the standard multipole expansion of the interaction operator. Substituting the bipolar expansion of 1 /r12 and analogous expansions for other terms of the operator V into the matrix elements (0 V J), the bipolar expansion of a given polarization correction is obtained. 

The bipolar expansion was first applied to interrnolecular interactions by Koide201 and by Linder et al.202,203. The electrostatic energy was considered in Ref. (204) while dispersion interactions were treated in Refs. (201-205) for two hydrogen atoms, and in Ref. (206) for many-electron atoms. Later, this work was extended to interactions of atoms with diatomics by Rosenkrantz and Krauss207 and to arbitrary systems by Knowles and Meath208. 

Buehler RJ, Hirschfelder JO (1951) Bipolar expansion of Coulombic potentials. Phys Rev 83 628-633... 

In the framework of the so called bipolar expansion of rnA [58], for two cylindrically symmetric, normalized charge distributions, the expression lob can be written as ... 

There are other approaches for calculations of the TERIs. The uniformly charged sphere model is used in LNDO/S, the multipole modeP is used in MNDO/d, AMl/d, and PM3(tm) the bipolar expansion of the Klopman-Ohno potential is used in PM3d. 

This integral equation can be solved by expansion of the integrand in bipolar coordmates [2, 3], Further improvement to the PY equation can be obtamed by analytical fit to simulation studies as described below. 

Bipolar disorders have been categorized into bipolar I disorder, bipolar II disorder, and bipolar disorder, not otherwise specified (NOS). Bipolar I disorder is characterized by one or more manic or mixed mood episodes. Bipolar II disorder is characterized by one or more major depressive episodes and at least one hypomanic episode. Hypomania is an abnormally and persistently elevated, expansive, or irritable mood, but not of sufficient severity to cause significant impairment in social or occupational function and does not require hospitalization. Most epidemiologic studies have looked at bipolar disorder of all types (bipolar I and bipolar II), or the bipolar spectrum, which includes all clinical conditions thought to be closely related to bipolar disorder. The lifetime prevalence of bipolar I disorder is estimated to be between 0.3% and 2.4%. The lifetime prevalence of bipolar II disorder ranges from 0.2% to 5%. When including the bipolar spectrum, the lifetime prevalence is between 3% and 6.5%.1... 

The diagnosis of mania is made on the basis of clinical history plus a mental state examination. Key features of mania include elevated, expansive or irritable mood accompanied by hyperactivity, pressure of speech, flight of ideas, grandiosity, hyposomnia and distractibility. Such episodes may alternate with severe depression, hence the term "bipolar illness", which is clinically similar to that seen in patients with "unipolar depression". In such cases, the mood can range from sadness to profound melancholia with feelings of guilt, anxiety, apprehension and suicidal ideation accompanied by anhedonia (lack of interest in work, food, sex, etc.). 

Mania. Mania and hypomania can also occur in children and adolescents on SSRIs, and, again, it is not known if there is an added developmental risk (Ven-kataraman et al., 1992). In a fluoxetine treatment study for depression, 3 (of 48) patients developed manic symptoms, even after excluding patients with psychotic depression, bipolar symptoms, or a family history of bipolar disorder (Emslie et al., 1997). In a paroxetine treatment study for depression, 5 adolescents (of 93) were removed for emotional lability and 1 for eupho-ria/expansive mood (Keller et al., 2001). 

Schizophrenia-related disorders, such as schizophreniform disorder, can closely mimic an acute exacerbation of mania. Attention to premorbid personal and family history may help differentiate them from mood disorders. A definitive diagnosis may not be possible, however, until the course of the illness is followed for a period of time. Clinical clues include the propensity of bipolar manics (in contrast to schizophrenics) to demonstrate pressured speech, flight of ideas, grandiosity, and overinclusive thinking. Hallucinations are less common than delusions in both mania and depression, with delusions normally taking on the qualities of expansivity, hyperreligiosity, or grandiosity. Delusions are also relatively less fixed than in schizophrenia. 

In the parametrization of equ. (4.68) the terms associated with the Legendre polynomials Pk(cos ab) represent that part of the angular correlation which is independent of the light beam, while the terms associated with the bipolar harmonics are due to the multipole expansion of the interactions of the electrons with the electric field vector. The link between geometrical angular functions and dynamical parameters is made by the summation indices ku k2 and k. These quantities are related to the orbital angular momenta of the two individual emitted electrons, and they are subject to the following conditions ... 

Fortune MT, Kennedy JL, Vincent JB. 2003. Anticipation and CAG CTG repeat expansion in schizophrenia and bipolar affective disorder. Curr Psychiatry Rep 5(2) 145-154. 

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