Expectation values of single-particle operators in the random phase approximation ground state

We developed a method for computing matrix elements of single-particle operators in the correlated random phase approximation ground state. Working with the explicit random phase approximation ground state wavefunction, we derived practically useful and simple expression for a molecular property in terms of random phase approximation amplitudes. The theory is illustrated by the calculation of molecular dipole moments for a set of representative molecules.Comment: Accepted to J.Chem.Phy

N-[(9E)-2-Chloro-9-thia-9H-xanthen-9-ylidene]-N-(4-fluorophenyl)amine

In the title compound, C19H11ClFNS, the central 4H-thiapyran ring of the 9H-thiaxanthene moiety shows a roof-shaped structure, with a dihedral angle of 34.3 (2)°. The molecules pack in the crystal structure via aromatic π-π interactions

5-Benzyl-1-(4-fluorophenyl)-2-phenyl-4,5,6,7-tetrahydro-1H-pyrrolo[3,2-c]pyridine

In the title compound, C26H23FN2, the dihedral angle between the 4-fluorophenyl ring and the adjacent phenyl ring is 62.3 (1)°. The crystal structure is stabilized by C-H...π interactions

1-(4-Chlorophenyl)-2,6,6-trimethyl-1,5,6,7-tetrahydro-4H-indol-4-one

In the title compound, C17H18ClNO, the tetrahydroindole ring system is nearly planar, except for the dimethyl-substituted C atom. Molecules are linked via C-H...O and C-H...π interactions, forming chains along the b axis

1-[1-(4-Fluorophenyl)-2-methyl-5-phenyl-1H-pyrrol-3-yl]ethanone

The title compound, C19H16FNO, crystallizes with two crystallographically independent molecules in the asymmetric unit. The dihedral angles between the pyrrole ring and fluorophenyl and unsubstituted phenyl rings are 44.9 (1) and 54.5 (2)°, respectively, in the first molecule, and 72.8 (3) and 30.7 (3)° in the second molecule. The crystal structure is stabilized by intermolecular C-H...O and C-H...π interactions

N-(3-Fluorophenyl)-9H-xanthen-9-ylideneamine

In the title compound, C19H12FNO, the dihedral angle between the mean planes of the 9H-xanthene moiety and the 3-fluorophenyl group is 82.5 (1)°. An intramolecular C-H interaction stabilizes the molecular conformation

trans-1,2-Difluoro-3,4,5,6,7,8-hexaphenyltricyclo[4.2.0.0^(2,5)]octa-3,7-diene

In order to probe the possible mechanism of the rearrangement of trans-hexaphenyldifluorotricyclooctadiene (a dimer of fluorotriphenylcyclobutadiene) to pentaphenyldihydrodifluoropentalene via C-F bond migration, a high-temperature study of the title compound, C_(44)H_(30)F_2, was performed at 400 (2) K. In the title compound, there are three fused four-membered rings with the resulting eight-membered tricyclo¬octa¬diene ring adopting a sofa conformation. The dihedral angles between the central four-membered ring and the two outer rings are 66.03 (2) and 65.39 (2)°. The crystal structure contains centrosymmetric dimers formed by C-H… π inter¬actions

N-Phenylethyl-N'-[3-(trifluoromethyl)phenyl]thiourea

The title compound, C16H15F3N2S, is a biologically active anti-implantation agent. The dihedral angle between the phenyl and trifluoromethylphenyl rings is 15.9 (2)°. The crystal structure is stabilized by intermolecular N-HS hydrogen bonds, forming dimers

4-(2-Methylprop-2-enyl)-1-[3-(trifluoromethyl)phenyl]thiosemicarbazide

The title compound, C12H14F3N3S, is a biologically active anti-implantation agent. Its crystal structure is stabilized by intermolecular N-H...S hydrogen bonds, which form dimers in a head-to-tail arrangement and link them into a polymeric chain

A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. The \emph{strong metric dimension}, $sdim(G)$, of a graph $G$ is the minimum among cardinalities of all strong resolving sets: $W \subseteq V(G)$ is a \emph{strong resolving set} of $G$ if for any $u, v \in V(G)$, there exists an $x \in W$ such that either $u$ lies on an $x-v$ geodesic or $v$ lies on an $x-u$ geodesic. In this paper, we prove that $Z(G) \le sdim(G)+3r(G)$ for a connected graph $G$, where $r(G)$ is the cycle rank of $G$. Further, we prove the sharp bound $Z(G) \leq sdim(G)$ when $G$ is a tree or a unicyclic graph, and we characterize trees $T$ attaining $Z(T)=sdim(T)$. It is easy to see that $sdim(T+e)-sdim(T)$ can be arbitrarily large for a tree $T$; we prove that $sdim(T+e) \ge sdim(T)-2$ and show that the bound is sharp.Comment: 8 pages, 5 figure