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科技英语写作经典练习（二）练习10I、将下列句子译成英语，要求把画线部分用名词从句来表示：1、现在有待于确定该级数（series）何时收敛（converge）。
2、从杜巴梅尔定理（Dubamel’s Theorem）可清楚地看出，这个极限（limit）是存在的。
3、这误差为多大，取决于几个因素。
4、采用（adopt）哪种观点（point of view）并不重要（a matter of indifference）。
5、应当强调指出，这些定义可用于任何大小（magnitude）的角度。
6、(作者)认为学生对机械制图（mechanical drawing）有了基本的了解。
7、由麦克斯韦假设（Maxwell’s hypothesis）得知，每当电场（electric field）发生变化时就产生出磁场（magnetic field）。
8、偶尔（it happens that）整个相位移（overall phase shift）为零。
9、在这种情况下磁铁（magnet）是否被运动是没有关系的。
10、选择哪个未知数（unknown）是没有什么区别的。
11、温度确定了热传递（transfer of heat）将朝哪个方向发生。
12、我们应该使用哪个式子取决于在题目中我们已知了什么数据。
13、一切物质（matter）均是由微粒构成的，这现在已是一个众所周知的事实了。
14、人们已经发现，电流的方向与电子流动的方向相反。
II、将下列句子译成英语，要求把画线部分用“what”从句来表示：1、必须确定在什么条件下这个方程是成立的。
2、这个设备正是我们所需要的。
3、现在我们把到目前为止所证明了的内容小结（summarize）一下。
4、发电机所做的是把机械能转变成电能。
5、这一章所讲的内容非常重要。
6、这个磁力（magnetic force）就是使电动机（electricmotor）转动（run）的力。
7、物质是能够占有空间的东西。
AIM – (Atoms In Molecule) An analysis method based upon the shape of the total electron density; used to define bonds, atoms, etc. Atomic charges computed using this theory are probably the most justifiable theoretically, but are often quite different from those from older analyses, such as Mulliken populations.AO – (Atomic Orbital) An orbital described by wavefunction for a single electron centered on a single atom.AOM – (Angular Overlap Model)AM1 – (Austin Model 1) Dewar's NDDO semiempirical parameterization.ASE – (Aromatic Stabilization Energy)BAC-MP4 –(Bond Additive Corrections to energies from Møller-Plesset 4th order perturbation theory) empirical corrections to calculated energy based (exponentially) on bond lengths.Bader's analysis – see AIMBasis Set –finite set of functions used to approximately express the Molecular Orbital wavefunction(s) of system, normally atom centered, consisting of AOs differing in local angular momentum for each atom.BCUT – (Burden CAS University of Texas) topological molecular similarity index of Burden. BDE – (Bond Dissociation Energy)BSSE – (Basis Set Superposition Error) error introduced when the energy of two molecules modeled together is lower than the sum of the energies when modeled separately, because more basis sets are available for each fragment (more degrees of freedom) during the calculation. Minimal basis sets can have less BSSE because only diffuse functions can span a to b. A particular problem for binding energies of weakly bonded molecular complexes, less with more complete basis sets.BLYP – (Becke-Lee-Yang-Parr)bohr - One atomic unit of distance, equal to 0.5292 Å.CADPAC – (Cambridge Analytical Derivatives PACkage)CAS – (Complete Active Space)CASPT – (Complete Active Space Perturbation Theory)CASSCF –(Complete Active Space Self-Consistent Field) popular variant of the MCSCF method, using a specific choice of configurations. One selects a set of active orbitals and active electrons, then forms all of the configurations possible by placing the active electrons in the active orbitals, consistent with the proper spin and space symmetry requirements. Essentially equivalent to the FORS method.CBS-QCI –(Complete Basis Set Quadratic Configuration Interaction) alternative extrapolation algorithm to complete basis set.CC –(Coupled Cluster) A perturbation theory of electron correlation with an excited configuration that is "coupled" to the reference configuration. Complete to infinite order, but only for a subset of possible excitations (doubles, for CCD). Newer than CI.CCD – (Coupled Cluster, Doubles only.) Complete to infinite order for doubles excitations. CCSD – (Coupled Cluster, Singles and Doubles only.) Complete to � order for singles & doubles excitations.CCSD(T) – (Coupled Cluster, Singles and Doubles with Triples treated approximately.) Size consistent. ΔHdiss ± 1.0 kcal/molCCSDT – (Coupled Cluster, Singles, Doubles and Triples)CCSS –(Correlated Capped Small Systems) Special case of IMOMO. See Truhlar's and Hass's work.CEPA – (Coupled Electron Pair Approximation)CFF – (Consistent Force Field) Class II force field based on ab initio data developed by Biosym Consortium/MSI.CFMM – (Continuous Fast Multipole Method) linear scaling method for matrix formation for DFT.CFSE – (Crystal Field Stabilization Energy)CHA – (Chemical Hamiltonian Approach) excludes BSSE effects from wavefunction w/ BSSE free Hamiltonian.CHF – (Coupled Hartree-Fock method)CHELP – (CHarges from Electrostatic Potential) an attempt to obtain atomic charges by fitting the electrostatic potential to a set of atomic point charges. The key component to theCHELP method is that it is non-iterative, rather using a Lagrangian multiplier method.chemical shift tensor –representation of the chemical shift. 3x3 matrix field felt in a direction induced by current due to applied field in b direction.CI – (Configuration Interaction) The simplest variational approach to incorporate dynamic electron correlation. Combination of the Hartree-Fock configuration (Slater determinant) and a large set of other configurations is used as a many e- basis set. The expansion coefficients are determined (in principal) by diagonalizing the Hamiltonian matrix and variationally minimizing the total energy of the CI wavefunction. Not size consistent.CID – (Configuration Interaction, Doubles substitution only) post-Hartree-Fock method that involves configuration interaction where the included configurations are the Hartree-Fock configuration and all configurations derived from doubles substitution.CIDEP – (Chemically Induced Dynamic Electron Polarization)CIDNP – (Chemically Induced Dynamic Nuclear Polarization)CIS – (Configuration Interaction with Singles excitations) simplest method for calculating electronically excited states; limited to singly-excited states. Contains no e- correlation and has no effect on the ground state (Hartree-Fock) energy.CISD –(Configuration Interaction, Singles and Doubles substitution only) post-Hartree-Fock method that involves configuration interaction where the included configurations are the Hartree-Fock configuration and all configuration derived from singles and doubles substitution. Comparable to MP2.CKFF – (Cotton-Kraihanzel Force Field)CNDO –(Complete Neglect of Differential Overlap) The simplest of the semi-empirical methods. The principle feature is the total neglect of overlap between different orbitals. In other words, the overlap matrix S is the unit matrix. The only two-electron integrals kept are those where electron 1 is in just one orbital and electron 2 is in just one orbital. Like all semiempirical methods the integrals are evaluated empirically.Combinatorial Chemistry – testing a large number of related compounds (as a mixture?) to find a compound that is active.CoMFA – (Comparative Molecular Field Analysis) technique used to establish 3-D similarity of molecular structure in relation to a target property.COMPASS –(Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies) Molecular Mechanics force field derived from Class II force field, but optimized forcondensed phase properties.contraction – the particular choice of scheme for generating the linear combinations of Gaussian functions that constitute a contracted basis set. A "generally-contracted" basis set is one in which each primitive is used in many basis functions. A "segmented" basis set, in contrast, is one in which each primitive is used in only one (or maybe two) contracted function.COOP – (Crystal Overlap pOPulation)Correspondence Principle –classical mechanics becomes a special case of quantum mechanics at larger masses/scale.COSMO – (Conductor-Like Screening Model) implicit solvation model of Andreas Klamt. Considers macroscopic dielectric continuum around solvent accessible surface of solute. Less sensitive to outlying charges then PCM.COSMO-RS – (COSMO for Realistic Solvents) describes solute and solvent on the same footing. Allows for realistic calculations of fluid thermodynamics: partition coefficients, solubilities, activity coefficients, vapor pressures, & phase diagrams.Coulomb integral –1-center, 2 electron integral in Hamiltonian. Represents localized contribution of e- pair (in A+B- or A-B+) to bonding.CP –(Counter Poise) correction method for BSSE based on assumption of additivity of effects caused by BSSE on intermolecular interactions. The BSSE is approximated as the energy difference between (1) an isolated fragment and (2) t fragment accompanied by the basis functions, but not the atoms, of its companion fragment(s).CPHF – (Coupled Perturbed Hartree-Fock )CVFF – (Consistent Valence Force Field)Density Matrix –DFT – (Density Functional Theory) ab initio electronic method from solid state physics. Tries to find best approximate functional to calculate energy from e- density. Scales as 2nd power times a large number. Static correlation built in. Not variational. Believed to be size consistent.diffuse –refers to basis functions that are typically of low angular momentum (unlike polarization functions) but with much smaller exponents, so that they spread more thinly over space. Usually essential for calculations involving negative ions or Rydberg states.DIIS – (Direct Inversion in the Iterative Space) Pulay's extremely efficient, extrapolation procedure, to accelerate the convergence of the SCF in Hartree-Fock calculations. DISCO –DOS – (Density Of States)dynamic correlation – All the correlation energy or correlation effect that is not considered "nondynamic" or "static." Essential for dispersion interactions.DZ – (Double-Zeta) A basis set for which there are twice as many basis functions as are minimally necessary. "Zeta" (Greek letter z) is the usual name for the exponent that characterizes a Gaussian function.DZP – (Double-Zeta with Polarization functions added) A polarization set generally has an angular momentum one unit higher than the highest valence function. So a polarization set on carbon is a set of d-functions.ECP = pseudopotential – (Effective Core Potential) The core electrons have been replaced by an effective potential. Saves computational expense. May sacrifice some accuracy, but can include some relativistic effects for heavy elements.EFFF – (Energy Factored Force Field)EHT – (Extended Hückel Theory)electron correlation – explicitly considering the effect of the interactions of specific e- pairs, rather than the effect each e- feels from the average of all the other e-. High correlation effects for e- rich systems, transition states, "unusual" coordination numbers, no unique Lewis structure, condensed multiple bonds, radicals, & biradicals.Embedded Cluster –Sauer's (and Teunissen's and others) approach to modeling large/extended systems, using a high accuracy (QM) method for the important site, and a less accurate (MM) method for the environment. Subtract out duplicated overlap term, in case of extended system. See also IMOMM, and CCSS.ESFF – (Extensible & Systematic Force Field) Rule based force field designed for generality rather than accuracy by Biosym/MSI.ESP – (ElectroStatic Potential) The electrical potential due to the nuclei and electrons in the molecule, as experienced by a test charge.exchange energy – Also called "exchange correlation energy." The energy associated with the correlation among the positions of electrons of like spin. This is included in Hartree-Fockcalculations.Exchange integral – integral in Hamiltonian centered on 2 atoms. Corresponds to covalent bond term.FMO – (Frontier Molecular Orbital Theory) reaction controlled by interaction and overlap (energy, symmetry, accessibility) of frontier MO. Fukui calculation of weighted contributions, but dominated by HOMO for electrophilic, LUMO for nucleophilic susceptibility, closest energy for free radical superdelocalizability.Fock Matrix – 1 particle/e- Hamiltonian; matrix of 1 e-, 2 e- Coulomb, & 2 e- Exchange integrals in total energyFock Operator – replaces the Hamiltonian Operator in the Schrödinger equation for a spin orbital/single electron in a mean field of the other electronsFORS – (Fully Optimized Reaction Space) The FORS method, first proposed by Ruedenberg and coworkers, is essentially equivalent to the CASSCF method, except for the implementation of the MCSCF.G1, G2 –(Gaussian 1 theory, Gaussian 2 theory) empirical algorithm to extrapolate to complete basis set and full correlation (beyond MP2/6-31G**) from combination of lower level calculations: HF/6-31G(d) frequencies; MP2/6-311G(dp) geometries; single point energies of MP4SDTQ w/ 6-311G**, 6-311+G** & 6-311G**(2df) and QCISD(T)/6-311G**. Practical up to ~7 heavy atoms. Cons: Cl, F BDE's. ΔHf ± 1.93 kcal/molG3 – (Gaussian 3 "slightly empirical" theory) extension of G2, add systematic correction for each paired e- (3.3 mHa) & each unpaired e- (3.1 mH a). ΔHf ± 1.45 kcal/molGaussians – functions frequently used as primitive functions to expand total wave function. Typically defined with Cartesian coordinates w/ respect to a point in space.GAPT – (Generalized Atomic Polar Tensor) A method for determining atomic charge on the basis of dipole moment derivatives.GGA –(Generalized Gradient Approximation) Corrects local density approximation by including a functional of the density gradient (i.e., f[grad rho]) in addition to the local functional. Favors greater densities and inhomogeneity, overcomes LDA tendency to overbind. Uses scale factors from density & 1st derivative of density. Adds ~20% to compute time.ghost function – A basis function that is not accompanied by an atomic nucleus, usually for counterpoise corrections for BSSE.GIAO –(Gauge Independent Atomic Orbitals) Ditchfield's method for canceling out the arbitrariness of the choice of origin & form (gauge) of the vector potential used to introduce the magnetic field in the Hamiltonian when calculating chemical shielding and chemical shift tensor. An exponential term containing the vector potential is included with each atomic orbital. Originally developed based on Hartree-Fock, improved by Pulay w/ DFT to be faster, also used w/MP2 & CCSD. Pros: less basis set dependence than IGAIMGreen's Function – inverse of the Hessian Matrix.GTO –(Gaussian-Type Orbital) Basis function consisting of a Gaussian function, i.e., exp(-r2), multiplied by an angular function. If the angular function is "Cartesian", there are six d-functions, ten f-functions, etc. (6d, 10f). If the angular function is spherical, there will the usual number of functions (5d, 7f).GUGA – (Graphical Unitary Group Approximation)GVB –(Generalized Valence Bond) Bill Goddard's "laid back Southern California wavefunctions." Equivalent to MCSCF correlation treatment: 2 configurations used to represent each e- pair and solved for 2 associated orthogonal functions. Excitations are taken within an electron pair but not between orbitals in different pairs Dissociation-consistent. If restricted to doubles, is called "perfect pairing" (GVB-PP). If includes both singles and doubles, is called "restricted configuration interaction" (GVB-RCI).Hamiltonian –Matrix of bare nucleus, 0 e- Fock Matrix. Matrix that operates on wavefunction to calculate the energy in Schrödingers equation.hartree – One atomic unit of energy, equal to 2625.5 kJ/mol, 627.5 kcal/mol, 27.211 eV, and 219474.6 cm-1.Hessian – Matrix of mass normalized 2nd derivatives of energy with respect to nuclear displacement in Cartesian coordinates. Diagonalized to determine normal modes & calculate vibrational frequencies.HF – (Hartree-Fock) Named after the developers of the most widely practiced method for solving the Schrödinger Equation for multi-electronic systems. The Hartree-Fock approximation (sometimes called the Self-Consistent Field Approximation) assumes a single (Slater) determinant wavefunction, or in other terms, the standard molecular orbital model. Approximates the molecular, all e- wavefunction as product of single e- wavefunctions, and approximate molecular wavefunctions as LCAO. Orbitals that contain e- are occupied, those that are vacant are called "virtual." The orbital for each electron is determined in the average field of the other electrons, and iterated until self-consistency is achieved. Uses exact form of the (electronic) Hamiltonian Operator in Schrödinger’s equation, and tries to find the best approximate wavefunction. Scales as n^3-4.HMO – (Hückel Molecular Orbital theory)HOMO – (Highest Occupied Molecular Orbital) The occupied molecular orbital having the greatest energy. In Frontier Molecular Orbital theory the HOMO plays a significant role in determining the course and barrier to reactions. The energy of this orbital approximates the ionization energy of the molecule by Koopman's Theorem.HONDO –IEHT – (Iterative Extended Hückel Theory)IEPA –(Independent Electron Pair Approximation) An alternative method to the CI approach for obtaining electron correlation. Assumes the correlation energy is pair-wise additive: the correlation energy for each pair of electrons is obtained independently of all other electrons and then summed up. The method was developed by Sinanoglu (who called it Many-Electron Theory or MET) and Nesbet (who called it Bethe-Goldstone Theory).IGLO –(Individual Gauge for Localized Orbitals) Kutzelnigg & Schiendler's method for canceling out the arbitrariness of the choice of origin & form (gauge) of the vector potential used to introduce the magnetic field in the Hamiltonian when calculating chemical shielding & chemical shift tensors at nuclei. An exponential term containing the vector potential premultiplies MOs that have been localized in real space. Faster than HF based GIAOIMOMM – (Integrated Molecular Orbital & Molecular Mechanics) Morokuma's method to extrapolate to high level calculation on large systems by integrating addition of ΔE between large system and small model systems calculated at low level (MM), to the energy of the small model calculated at a high level (MO).IMOMO – (Integrated Molecular Orbital & Molecular Orbital) same as IMOMM with high (ab initio) and low level (semiempirical) MO calculations.INDO –(Intermediate Neglect of Differential Overlap) A semiempirical method closely related to the CNDO method, where all terms of the Fock matrix used in CNDO are included, but the restriction that the monocentric two-electron integrals all be equal is lifted. internal coordinates –Bond lengths, bond angles, and dihedral (torsional) angles; sometimes called "natural coordinates."IRC – (Intrinsic Reaction Coordinate) An optimized reaction path that is followed downhill, starting from a transition state, to approximate the course (mechanism) of an elementary reaction step. (Ignores tunneling, contribution of vibrationally excited modes/partition function, etc.)isodesmic –a chemical reaction that conserves types of chemical bond. Due to bettercancellation of systematic errors, energy changes computed using such reactions are expected to be more accurate than those computed using reactions that do not conserve bond types.isogyric –a chemical reaction that conserves net spin. Due to better cancellation of systematic errors, energy changes computed using such reactions are expected to be more accurate than those computed using reactions that do not conserve spin.Koopman's Theorem – the ionization potential can be approximated by the energy of the HOMO. Errors from neglect of e- correlation & e- relaxation tend to cancel for IP, but compound for trying to approximate EA from LUMO energy.LCAO – (Linear Combination of Atomic Orbitals) Molecular orbitals are usually constructed as a linear combination of atomic orbitals. The coefficients in this expansion are determined by solving the Schrödinger equation, typically in the Hartree-Fock approximation.LDA = LSDA – (Local Density/Spin-Density Approximation) A DFT method involving only local functionals: those that depend only upon the value of the density, f[rho]: no dependence upon the gradient of the electron density.level shifting – method to avoid oscillations during SCF convergence by artificially raising the energies of the virtual orbitals to separate the different states.LFER – (Linear Free Energy Relationship)LNDO – (Local Neglect of Differential Overlap)LORG – (Localized Orbitals, Local Origin) Hansen & Bouman Gauge invariant method for calculating chemical shielding & chemical shift tensors: expanding angular momentum terms relative to local origin for each orbital so no reference to gauge origin. Similar to IGLO.LST – (Linear Synchronous Transit) An interpolative method to provide an initial guess for a transition state. Assumes each atom position in the transition state is in a direct line between its position in the reactants and its position in the products.LUMO – (Lowest Unoccupied Molecular Orbital) The unoccupied molecular orbital having the lowest energy. In Frontier Molecular Orbital theory the LUMO plays a significant role in determining the course and barrier to reactions. The energy of the LUMO is a poor approximation of the EA.Madelung Potential – a set of point charges used to represent a solid continuum.MBPT = Møller-Plesset Perturbation Theory – (Many Body Perturbation Theory)MC – (Monte Carlo) method of sampling based on generating random numbers.MCCM – (Multi Coefficient Correction Method) Truhlar's method to scale each component of the correlation energy to extrapolate to full correlation: HF, MP2, and MP4 terms. MCPF – (Modified Couple Pair Functional)MCSCF – (MultiConfiguration Self-Consistent Field) A limited type of CI. A simple extension of the SCF approach to include non-dynamic electron correlation, with the added feature of orbital optimization. A number of configurations are chosen in some manner, then both the expansion coefficients and the orbital coefficients are optimized.MD – (Molecular Dynamics) method of sampling geometries based on Newtonian laws of motion.MEP – (Molecular Electrostatic Potential)MERP – (Minimum Energy Reaction Path)MINDO –(Modified Intermediate Neglect of Differential Overlap) Refers to the parametrization set (and computer code) developed by the Dewar group for performing INDO calculations. The most useful of the MINDO series is MINDO/3.Minimal Basis Set – The smallest # of functions needed to hold all electrons and still be spherical. Commonly refers to STO-3G.MM – (Molecular Mechanics) classical description of molecules as atoms held together by spring-like bonds. Chemical "Hamiltonian" based on force constants.MM2 –(Molecular Mechanics, Allinger Force Field version 2) one of the earliest, and probably the best known and tested Molecular Mechanics force field for organic molecules. MMFF – (Merck Molecular Force Field)MMP2 –(Molecular Mechanics, Allinger Force Field version 2 with Pi electrons handled quantum mechanically). Pi bond stretch calculated on the fly based on simple SCF bond order.MM3 – (Molecular Mechanics, Allinger Force Field version 3)MNDO –(Modified Neglect of Diatomic Overlap) Dewar’s first parametrization of the NDDO semiempirical method. First incorporated into the MOPAC program.MNDOC –(Modified Neglect of Diatomic Overlap Correlated) Thiel’s version of MNDO whichincludes limited configuration interaction.MNDO/d – (MNDO with d-orbitals) Thiel's version of MNDO with extra set of orbitals for second row transition and main group elements.MO – (Molecular Orbital) An orbital described by the wavefunction for a single electron in a molecule, usually delocalized across the entire molecule. Usually, molecular orbitals are constructed as combinations of atomic orbitals.MP2 –(Møller-Plesset theory, 2nd order) Møller-Plesset developed the perturbative approximation to include electron correlation, using the Hartree-Fock wavefunction as the zeroth order wavefunction. MP2 refers to the second order energy correction. Comparable to CISD.MP3 –(Møller-Plesset Theory, 3rd order) A perturbative approximation for including electron correlation, using the Hartree-Fock wavefunction as the zeroth order wavefunction. MP3 refers to the third-order energy correction.MP4 –(Møller-Plesset Theory, 4th order) A perturbative approximation for including electron correlation, using the Hartree-Fock wavefunction as the zeroth order wavefunction. MP4 refers to the fourth order energy correction. MP4 as implemented in GAUSSIAN can be calculated using various configurations. For example, MP4SDQ means that all terms involving singles, doubles and quadruples configurations are included through fourth order.MPA – ( Mulliken Population Analysis)MR – (Multi-Reference)MRCI –(Multi-Reference Configuration Interaction) CI using more that one reference determinant, instead of the usual single Hartree-Fock reference (ie. the ground state wavefunction). Among multi-reference theories, MR-CISD (singles and doubles CI) is popular and high-level (but not dissociation consistent).Mulliken Charges – Charges are assigned to each atom from an electronic calculation, by arbitrarily attributing the e- density of the diagonal terms (single atom terms) to each atom, plus half of the e- density of each off-diagonal term for that atom. Very sensitive (<100%) to basis set.NAO – (Natural Atomic Orbital)natural orbital – those orbitals for which the first-order density matrix is diagonal; each will contain some non-integer number of electrons between 0 and 2. Usually discussed in the context of a correlated calculation. RHF calculations give molecular orbitals that are also natural orbitals. The NOs are the orbitals for which the CI expansion converges fastest.Particularly valuable in modeling organometallic complexes.NEMD – (Non Equilibrium Molecular Dynamics) MD w/ applied forces, such as shear. NBO –NDDO – (Neglect of Differential Diatomic Overlap) This semiempirical method that keeps all terms of the Fock matrix except those involving diatomic differential overlap: the only two-electron terms, are those where e- 1 is in one orbital on atom A and e- 2 is in one orbital on atom B.NPA – (Natural Population Analysis) A method for determining atomic charge distributions in a molecule. Based on creating atomic natural orbitals from the molecular orbitals, by finding eigenfunctions of atomic sub-blocks of the molecular density matrix, based on chemical concepts of bonds, lone pairs, etc. Fairly independent of basis set.OPLS – (Optimized Potentials for Liquid Simulations) force field for condensed (aqueous) simulations. Electrostatic only for H-bond, "Disappearing L-J" H-bond.ONIOM – (Our own N-layered Integrated molecular Orbital molecular Mechanics method) Morokuma's generalized method for modeling large/condensed systems. Model critical part of chemical system with a high accurate/level method, intermediate region with a less expensive method, outer region approximately.orbital – the wavefunction describing where an e- is in an atom or molecules. Usually an eigenfunction of a one-electron Hamiltonian, e.g., from Hartree-Fock theory. A spin orbital has an explicit spin and a spatial orbital does not. Orbitals are probably the most useful concept from quantum chemistry: one can think of an atom or molecule as having a set of orbitals that are filled with electrons (occupied) or vacant (unoccupied or "virtual".Pauling point – calculation that gives a good result because systematic, but opposed errors cancel out.PCI –(Parameterized Configuration Interaction) empirical scaling of contribution of dynamic correlation to bond strengths. See Siegbahn, Blomberg, et. al.PCM – (Polar Continuum Model) implicit solvation model of Tomasi.PES –(Potential Energy Surface) The 3N-6 (or 3N-5, for linear molecules) dimensional function that indicates how the molecule's energy depends upon its geometry. Perturbation Theory – approximation to include electron correlation, based on T aylor Series expansion of the Hartree-Fock wavefunction (1 e- Hamiltonian) as the 0th orderwavefunction, for E in Schrödinger equation, truncated after n terms.PM3 – (Parametric Method number 3) A re-parameterized version of AM1 by Stewart. PMO – (Perturbation Molecular Orbital theory)PNDO – (Partial Neglect of Differential Overlap)polarized – basis set that includes functions that are of higher angular momentum than is minimally required. (Combination of p-orbitals added to the valence s-orbitals, or d-orbitals in addition to valence p-orbitals on carbon, for instance.) The added functions are often called "polarization functions." Polarization functions help to account for the fact that atoms within molecules are not spherical.PPP – (Pariser-Parr-Pople)PRDDO – (Partial Retention of Diatomic Differential Overlap)primitive – individual Gaussian functions used singly to produce a contracted basis function: a set of p-functions is three basis functions, but may be many primitives (3n, where there are n primitives in the cGTO).PRISM – (Polymer Reference Interaction Site Model) Adaptation of RISM theory for liquid polymers by Schweizer & Curro. Based on single chain conformational statistics and interaction potentials.PUHF – (spin-Projected UHF) An approximation to provide the energy that would result from a UHF calculation, if it did not suffer spin-contamination. The PUHF energy is usually lower than the UHF energy because the contributions of higher-multiplicity states, which usually have high energies, have been (approximately) subtracted.QCI –(Quadratic Configuration Interaction) CI method with terms added to confer size-consistency. An approximation to coupled-cluster theory.QCISD(T) – (Quadratic Configuration Interaction with all Single and Double excitations and perturbative inclusion of Triple excitations) post Hartree-Fock method. Similar results to QCISDT, used in G2. Scales as n^7.QCPE –(Quantum Chemistry Program Exchange) The Quantum Chemistry Program Exchange is a warehouse of chemistry computer codes. They have operated since 1962, providing a depository of what is now well over 600 different programs. QCPE can be contacted at QCPE Creative Arts Building 181 Indiana University Bloomington, IN 47405 USA telephone: (812)855-4784 fax: (812)855-5539 E-mail: ****************.edu anonymous ftp access: 。
Supporting InformationStable electronic structures of a defective uranofullereneXing Dai a, Minsi Xin a, Yan Meng a, b, Jie Han a, Yang Gao a, Wei Zhang c, Mingxing Jin a, Zhigang Wang a, d,*, Rui-Qin Zhang b, d,*a Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, Chinab Department of Physics and Materials Science and Centre for Functional Photonics (CFP), City University of Hong Kong, Hong Kong SAR, Chinac Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, P. R. Chinad Beijing Computational Science Research Center, Beijing 100084, P. R. China*To whom correspondence should be addressed. E-mail: wangzg@, aprqz@.hk1. Three kinds of defective fullerenes: C61-Def[5, 6], C60 and C61-Def[6, 6].2. The isolate U2 molecule.3. Spin-orbit coupling effects for U2@C61.4. The EMF U2@C60.5. Six isomers of U2@C61-Def[5, 6].6. Charge distribution on two special carbon atoms.7. Characteristic molecular orbitals of U2@C60 and U2@C61-Def[6, 6].1.Three kinds of defective fullerenes: C61-Def[5, 6], C60 and C61-Def[6, 6].C60has two types of C-C bonds, i.e., [5, 6] and [6, 6] bonds. Previous work had studied the defective C61-Def[6, 6] fullerene formed by adsorbing a C atom on a [6, 6] bond of C60 [26]. In this work, we performed related calculations on the C61-def[5, 6] fullerene. The stable geometry structures of a C61-Def[5, 6], a perfect C60 and a C61-Def[6, 6] fullerenes are presented in Figure S1. Compared with the perfect C60, both the two types of defective fullerenes present enlarged local sizes of their cavities.Fig. S1. (a), (b) and (c) represent the equilibrium geometry structures of a C61-Def[5, 6], a perfect C60 and a C61-Def[6, 6] fullerenes.The ground states of the two defective fullerenes are summarized in Table S1. It is seen that the ground states of the two defective fullerenes are all triplet, due to the dangling bond of the adatom. Similar effects also exist in other carbon materials of graphene and nanotubes. In addition, compared with the C61-Def[6, 6], the C61-Def[5, 6] fullerene has a lower total energy and thus is more stable than the former.Figure S2 depicts the spin densities of the ground states of these three fullerene from what we can clearly realize the origins of the spin state. In the two types of defective fullerenes, thevast majority of net spin electrons are distributed on the adatom (see Figure S1(a) and (c)). From the data of Table S1, the net spin electrons of the adatom of the C61-Def[5, 6] and the C61-Def[6, 6] are about 1.4 and 1.5, respectively. A small amount of net spin electrons also exist on the two fullerene cages. The perfect C60 fullerene has a singlet ground state and there is no net spin on any atom. An interesting charge distribution is that there are a small amount of positive charge (about 0.1) on the adatom in C61-Def[5, 6]; however, there are almost no charge on the adatom in the C61-Def[6, 6] (see Table S1).Table S1. Electronic states of the two defective fullerenes, together with the number of net spin electrons and Mulliken charges on the adatom.Method Systems El. State ΔE(eV) Net spinon the adatomMulliken chargeon the adatomBP86 C61Def[5, 6] 1A’0.043A’0 1.39 0.10 5A’ 1.50C61Def[6, 6] 1A10.363B20.22 1.54 0.015A11.59PBE C61Def[5, 6] 1A’0.073A’0 1.40 0.10 5A’ 1.50C611A10.36Def[6, 6] 3B20.22 1.56 0.025A1.591Fig. S2. (a), (b) and (c) present the spin density of C61-Def[5, 6], C60and C61-Def[6, 6], respectively. The blue areas correspond to the spin up electrons, and the green areas show the spin down electrons. (Isovalue=0.002 a. u.)2. The isolate U2 molecule.According to the high-level quantum ab initio result [35], the isolate U2 molecule has a septet ground state with a U-U bond length of 2.43 Å. We performed a geometry optimization calculation for the U2 molecule as a test. From Table S2, the XC-functionals of both BP86 and PBE predict correct results of the ground state of the U2. Although the bond length is shorter than that obtained by CASSCF/CASPT2 calculations, it agrees well with previous DFT results [19]. Table S2. Electronic states and bond length of the isolate U2 molecule.System Method Electronic State ΔE (eV)R U-U (Å)U2BP86 Quintet 0.11Septet 0 2.27Nonet 0.23PBE Quintet 0.13Septet 0 2.26Nonet 0.253. Spin-orbit coupling effects for U2@C61.We considered spin-orbit coupling effects for U2@C61 and performed more detailed calculations and more systematic analysis to verify the reliability of our conclusions. Here, we employed the most accurate methods in Gaussian 09, Dmol3[S1] and ADF 2012.01[S2] to recalculate the structure (b). All of the results are similar with what we report in the manuscript.(1) DFT/ECP calculations.The method can consider the relativistic effect by using the relativistic effective core potential (RECP) performed in Gaussian 09 program, but not include spin-orbit effect. In spite of this, the electronic structure of U@C82 has been explained successfully by using this method [18] and confirmed the experimental observation [15]. More details about this method have been described in the calculation method section in the manuscript. Here, together with the relevant data of U2@C61-def[5, 6] (structure (b) in Figure 1) in Table S3, these results indicate that U2@C61-def[5, 6] has a quintet ground.Table S3. DFT/ECP results.U2@C61-def[5, 6] (b) Multiplicity ΔE (eV)BP86/SEG~3-21G Triplet 0.071Quintet 0Septet 0.005PBE/SEG~3-21G Triplet 0.137Quintet 0Septet 0.007(2). DKH4-DFT (DFT = BP86 and PBE) calculations.Further, we calculated the single-point energies for each structure obtained from method (1) using the DKH4-DFT method and compared their total energies at different multiplicities. In this method, the DKH4 Hamiltonian was employed, the all-electron scalar relativistic basis sets, SARC-DKH [S3] and cc-pvdz-DK [S4], were chosen for U atoms and C atoms, respectively. It is the most accurate method including relativistic effect available in Gaussian 09 program which is different from method (1), but cannot consider the spin-orbit effect either. When performing geometry optimization calculations using DKH4 method, the number of the total variables is limited within 50, so we only calculated the single-point energies for each structure. The results are summarized in Table S4 and we can see that the quintet of structure (b) still has the lowest total energy. Herein, although the energy difference between quintet and septet becomes smaller, about 0.002 eV, which may result from the structure that can’t relax sufficiently, it does not change the essential conclusion. In fact, the total energies of quintet and septet are indeed very closed (see below).Table S4. DKH4-DFT results.U2@C61-def[5, 6] (b) Multiplicity ΔE (eV)DKH4-BP86 (Single-Point) Triplet 0.115 Quintet 0 Septet 0.002DKH4-PBE (Single-Point) Triplet 0.170 Quintet 0 Septet 0.002(3) All-electron relativistic DFT/DNP (DFT = BP and PBE) calculationsIn the theoretical study of U2@C60 [18], the author used an all-electron scalar relativistic method based on the Douglas-Kroll-Hess (DKH) Hamiltonian to consider the relativistic effect. The PBE functional and the all-electron double-numerical basis set with polarization functions (DNP) were employed. This is the most accurate approach available in DMol3 package. In spite of the exclusion of the spin-orbit effect, the author expound that this method is capable of predicting the bonding scheme and the electronic states of U2@C60 qualitatively. Therefore, we employed the same method for U2@C61-def[5, 6] and re-optimized the structure (b) using Dmol3 package and listed the results in Table S5. The changes of the structure are very small compared to method (1). Table S5 also showed that the ground state of U2@C61-def[5, 6] is quintet.Table S5. All-electron relativistic DFT/DNP results.U2@C61-def[5, 6] (b) Multiplicity ΔE (eV)BP/DNP Triplet 0.164(Optimization) Quintet 0Septet 0.008PBE/DNP (Optimization) Triplet 0.173 Quintet 0 Septet 0.010(4) Calculations using acalar and Spin-orbit Relativistic DFT/TZP method via ZORA model.Both scalar and spin-orbit relativistic effects could be demonstrated by calculations using ADF program based on ZORA model. In previous EMFs studies containing actinides [S5], to achieve good calculation efficiency, the 1s-5d atomic orbitals of uranium and the 1s atomic orbitals of carbons were frozen (corresponding to 14 valence electrons for U and 4 valence electrons for C). Here, we chose the TZP Slater basis sets (relativistic valence triple-zeta with one polarization function) for all atoms. The 1s-4d atomic orbitals of U atom were frozen (corresponding to 32 valence electrons). All the atomic orbitals of C atoms were treated as valence orbitals. This approach is more accurate and undoubtedly more time-consuming. Firstly, we re-optimized the geometry of U2@C61-def[5, 6], and then calculated the single-point energies for the obtained structures with the spin-orbit effects considered. The qualitative results listed in Table S6 show that the quintet spin ground state does not change after including the spin-orbit coupling correction.Table S6. Results using scalar and Spin-orbit Relativistic DFT/TZP method based on ZORA model.U2@C61-def[5, 6] (b) Multiplicity ΔE (eV)BP/TZP (Scalar Relativistic) (Optimization) Triplet 0.328 Quintet 0 Septet 0.083BP/TZP (Spin-orbit Relativistic) (Single Point) Triplet 0.078 Quintet 0 Septet 0.044PBE/TZP (Scalar Relativistic) (Optimization) Triplet 0.318 Quintet 0 Septet 0.017PBE/TZP (Spin-orbit Relativistic) (Single Point) Triplet 0.075 Quintet 0 Septet 0.006Based on the discussion above, we employed a variety of methods to consider relativistic and spin-orbit effects and all the results revealed that U2@C61-def[5, 6] has a quintet ground state. Therefore we can conclude that the electronic states we obtained in the manuscript is qualitatively correct.4.The EMF U2@C60.In Table S7, the two methods present consistent predictions of the ground states and equilibrium geometry structures for the U2@C60 (mainly refer to the U-U distance). This system has a septet ground state with a U-U distance of about 2.71 Å. This distance value is consistent with previous report of 2.72 Å.Table S7. Electronic states, HOMO-LUMO gap and U-U distance of the U2@C60.Method System El. State ΔE (eV)Gap (eV) R U-U (Å)BP86 U2@C605A u0.067Au0 0.12 2.719Ag0.48PBE U2@C605A u0.107Bu0 0.13 2.719Ag0.505. Six isomers of U2@C61-Def[5, 6].Due to the different adsorption sites of the adatom, the U2@C61-Def[5, 6] has six isomers which are defined as structures (a) to (f) in Figure 1 of the main paper. We performed full geometry optimization calculations for the six isomers at different multiplicities and the total energy data are summarized in Table S8. From Table S8, the isomer (b) has the lowest total energy at quintet. So, the isomer (b) can be seen as the most stable U2@C61-Def[5, 6] and its related properties are discussed in the main paper.Table S8. Total energies of the six isomers of the U2@C61-Def[5, 6] at different multiplicities in BP86 calculations.Isomers Multiplicities ΔE (eV)[5, 6] (a) Triplet 0.16 Quintet 0.15Septet 0.16[5, 6] (b) Triplet 0.07 Quintet 0 Septet 0.01 Nonet 0.04[5, 6] (c) Quintet 0.19 Septet 0.18 Nonet 0.32[5, 6] (d) Singlet 0.42 Triplet 0.09 Quintet 0.37 Septet 0.38 Nonet 0.56[5, 6] (e) Quintet 0.12 Septet 0.10 Nonet 0.27[5, 6] (f) Quintet 0.06 Septet 0.04 Nonet 0.236. Charge distribution on two special carbon atoms.As mentioned in the manuscript, two special carbon atoms of the U2@C61-Def[6, 6] bonding with the adatom take the most negative charge among all carbon atoms due to the acceptance ofelectrons both from the adatom and the internal metal. In this section, we summarized the charge number of the two special carbon atoms of each fullerene in Table S9. The two carbon atoms are named as C1 and C2.Table S9. Charge distribution on the two special carbon atoms which bond with the adatom.BP86 PBEC1 C2 C1 C2 C600 0 0 0C61-Def[6, 6] -0.10 -0.10 -0.10 -0.10C61-Def[5, 6] -0.13 -0.13 -0.14 -0.14U2@C60-0.16 -0.16 -0.15 -0.15U2@C61-Def[6, 6] -0.28 -0.28 -0.29 -0.29U2@C61-Def[5, 6] -0.08 -0.15 -0.08 -0.16It is noted that in the U2@C60, each carbon atom of a six-member ring facing to a U atom takes almost equal negative charges (about - 0.16). In the U2@C61-Def[6, 6], each of the two special carbon atoms of C1 and C2 takes about - 0.28 negative charges, while the charge number range of other carbon atoms is about - 0.01 to - 0.21. In the U2@C61-Def[5, 6], the charge numbers of C1 and C2 are about -0.08 and -0.15, respectively. Most of the negative charge of the U2@C61-Def[5, 6] cage distributes at the two hexagons of the two poles (about -0.13 to -0.16 for each carbon atom).7. Characteristic molecular orbitals of U2@C60 and U2@C61-Def[6, 6].Figure S3 presents the characteristic MOs which can clearly reflect the U-U and U-cage covalent interaction in the U2@C60 system. In Figure S3(a), six single electron occupied MOs are almost completely dominated by 5f orbitals of the U atom. These orbitals indicate that there is a unique six-fold single electron U-U covalent bond in the C60 fullerene. Four double electron occupied MOs with lower energy in Figure S3 (b) have the characteristics of hybridization between the metal and the cage and indicate the covalent interaction. These results are consistent with Wu’s report [18].Fig. S3. (a) Six single electron occupied MOs reflect the six-fold single electron U-U covalent bond in the C60fullerene. And (b) four double electron occupied MOs reflect the covalent interaction between the metal and the cage. (Isovalue=0.035 a. u.)Figure S4 shows the corresponding MOs of the U2@C61-Def[6, 6]. Similar to the U2@C61-Def[5, 6] discussed in the manuscript, five single electron occupied MOs dominated by 5f orbitals of the U atom reflecting the U-U covalent interaction were found (see Figure S4(a)). Some deformations of these orbitals caused by the impact of the defect occur. Orbitals hybridization characters reflect the covalent U-cage interaction (see Figure S4(b)). These results agree with our previous report [26].Fig. S4. (a) Five single electron occupied MOs reflect the six-fold single electron U-U covalent bond in the C61-def[6, 6] fullerene. And (b) four double electron occupied MOs reflect the covalent interaction between the metal and the cage. (Isovalue=0.035 a. u.)According to the analysis on the characteristic MOs we can conclude that the two U atoms are binding by a covalent interaction which is dominated by 5f orbitals of the U atoms in the three types of fullerenes. Both the electrostatic attraction between the metal and the cage caused by the charge transfer and the hybridization characters between the two parts reflect the covalent U-cage interactions. Therefore, in these three EMFs, the essence of the interaction between the metal and cage is a mixture of ionic and covalent interaction.References[S1] Delley B. From molecules to solids with the DMol3 approach. J. Chem. Phys. 2000; 113:7756-7764. DMol3 is available as part of Materials Studio.[S2] ADF 2012.01 S, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, .[S3] Pantazis DA, Neese F, All-electron scalar relativistic basis sets for the actinides. J Chem Theory Comput 2011; 7(3): 677-684.[S4] Dunning Jr TH, Gaussian basis sets for use in correlated molecular calculation. I. The atoms boron through neon and hydrogen. J Chem Phys 1989; 90(2): 1007-1023.[S5] Dognon J-P, Clavaguera C, Pyykko P. A Predicted Organometallic Series Following a32-Electron Principle: An@C28 (An = Th, Pa+, U2+, Pu4+). J Am Chem Soc 2009; 131: 238-243.。
a r X i v :n u c l -t h /0311081v 1 21 N o v 2003Relativistic Green’s function approach to charged-current neutrino-nucleusquasielastic scatteringAndrea Meucci,Carlotta Giusti,and Franco Davide PacatiDipartimento di Fisica Nucleare e Teorica,Universit`a degli Studi di Pavia and Istituto Nazionale di Fisica Nucleare,Sezione di Pavia,I-27100Pavia,Italy(Dated:February 8,2008)A relativistic Green’s function approach to inclusive quasielastic charged-current neutrino-nucleus scattering is developed.The components of the hadron tensor are written in terms of the single-particle Green’s function,which is expanded on the eigenfunctions of the nuclear optical potential,so that final state interactions are accounted for by means of a complex optical potential but without a loss of flux.Results for the (νµ,µ−)reaction on 16O and 12C target nuclei are presented and discussed.A reasonable agreement of the flux-averaged cross section on 12C with experimental data is achieved.PACS numbers:25.30.Pt:Neutrino scattering,13.15.+g:Neutrino interactions,24.10.Jv:Relativistic mod-els,:Many-body theoryI.INTRODUCTIONThe reactions with an incident neutrino interacting with a complex nucleus have gained in recent years a wide interest,owing both to astrophysical reasons and to the aim of investigating the neutrino properties with a high accuracy.Besides the measurements with large underground detectors,some experiments have also been performed [1,2,3]using a pion beam which weakly de-cays producing leptons.In this case the most part of the neutrinos which are obtained is related to muons,with a smaller component of electron neutrinos.Both weak neutral-and charged-current scattering have stimulated detailed investigations.In particular,we are here interested in charged-current reactions at an energy below 1GeV as they have shown to be depen-dent on nuclear structure effects.Different approaches have been applied to investigate such processes,includ-ing the so-called “elementary particle model”[4],random phase approximation (RPA)in the framework of a rela-tivistic Fermi gas model [5]or a Fermi gas model with local density approximation [6],shell model [7,8]and relativistic shell model [9],RPA among quasiparticles [8]and continuum RPA [10,11].The reaction goes through a quasielastic (QE)mechanism,where the neutrino inter-acts with one single neutron and a proton together with a negative muon are emitted.The effect of final state interactions (FSI)has been stressed to significantly con-tribute to the cross section [12]and has been calculated with a relativistic shell model including a phenomenolog-ical optical potential,which describes the interaction of the outgoing nucleon with the residual nucleus,with and without imaginary part [13].The optical potential is fitted to reproduce the elas-tic proton-nucleus scattering through its real component,while its imaginary part takes into account the scatter-ing towards the inelastic channels.This means that the reaction channels are globally described by a loss of flux produced by the imaginary part of the complex poten-tial.This model has been applied with great success to exclusive QE electron scattering [14],where it is able to explain the experimental cross sections of one-nucleon knockout reactions in a range of nuclei from carbon to lead.In an inclusive process,however,where some of the reaction products are not detected and the inelastic channels are also included in the experimental cross sec-tion,the flux must be conserved.This fact is sometimes described by dropping the imaginary part of the optical potential.This procedure conserves the flux but it is not consistent with the exclusive reaction,which can only be reproduced with a careful treatment of the optical poten-tial,including both real and imaginary parts [14].In this paper we apply a Green’s function approach where the conservation of flux is preserved and FSI are treated in the inclusive reaction consistently with the ex-clusive one.This method was discussed in a nonrelativis-tic [15]and in a relativistic [16]framework for the case of inclusive electron scattering and it is here adapted,in a relativistic framework,to charged-current neutrino scattering.In this approach,the components of the nu-clear response are written in terms of the single-particle optical-model Green’s function.This result was origi-nally derived by arguments based on the multiple scatter-ing theory [17]and successively by means of the Feshbach projection operator formalism [15,18,19,20].The spec-tral representation of the single-particle Green’s function,based on a biorthogonal expansion in terms of the eigen-functions of the non-Hermitian optical potential,allows one to perform explicit calculations and to treat FSI con-sistently in the inclusive and in the exclusive reactions.Important and peculiar effects are given in the inclusive (e,e ′)reaction by the imaginary part of the optical po-tential,which is responsible for the redistribution of the strength among different channels.In Sec.II the general formalism of the charged-current neutrino-nucleus scattering is given.In Sec.III,the Green’s function approach is briefly reviewed.In Sec.IV,the results obtained on 16O and 12C target nuclei are2presented and discussed.Some conclusions are drawn in Sec.V.II.THE INCLUSIVE CROSS SECTIONIn a charged-current process neutrinos and antineutri-nos interact with nuclei via the exchange of weak-vector bosons and charged leptons are produced in the final state.The cross section of an inclusive reaction where an incident neutrino or antineutrino,with four-momentum k µi =(εi ,k i ),is absorbed by a nucleus and only the out-going lepton,with four-momentum k µ=(ε,k )and mass m ,is detected,is given by the contraction between the lepton tensor and the hadron tensor,i.e.,d σ=G 2cos 2ϑc(2π)3,(1)where G ≃1.16639×10−11MeV −2is the Fermi constant and ϑc is the Cabibbo angle (cos ϑc ≃0.9749).The lepton tensor isL µν=1ε spin¯u f γµ(1∓γ5)u i ¯u i (1±γ5)γνu f ,(2)where the upper (lower)sign corresponds to neutrino (an-tineutrino)scattering.After projecting into the initialneutrino (antineutrino)and the final lepton state,one hasL µν=1εi ε[l µνS ∓l µνA ],(4)withl µνS =k µi k ν+k νi k µ−g µνk i ·kl µνA =i ǫµναβk iαk β,(5)where ǫµναβis the antisymmetric tensor with ǫ0123=−ǫ0123=1.Assuming the reference frame where the z -axis is par-allel to the momentum transfer q =k i −k and the y -axis is parallel to k i ×k ,the symmetrical components l 0y S ,l xy S ,l zy S ,and the antisymmetrical ones l 0x A ,l xz A ,l 0z A ,as well as those obtained from them by exchanging their indices,vanish.The hadron tensor is given by bilinear products of the transition matrix elements of the nuclear weak charged-current operator J µbetween the initial state |Ψ0 of the nucleus,of energy E 0,and the final states |Ψf ,of energy E f ,both eigenstates of the (A +1)-body Hamiltonian H ,asW µν(ω,q )=fΨf |J µ(q )|Ψ0 × Ψ0|J ν†(q )|Ψf δ(E 0+ω−E f ).(6)and involves an average over initial states and a sum over the undetected final states.The sum runs over the scat-tering states corresponding to all of the allowed asymp-totic configurations and includes possible discrete states.The transition matrix elements are calculated in the first order perturbation theory and in the impulse ap-proximation,i.e.,the incident neutrino interacts with only one nucleon while the other ones behave as specta-tors.The current operator is assumed to be adequately described as the sum of single-nucleon currents,corre-sponding to the weak charged currentj µ=F V 1(Q 2)γµ+iκ(1+Q 2/M 2A)2,(8)F P =2MG Ad εdΩ=kεG 23The coefficients v ,obtained from the lepton tensor com-ponents,arev 0=1+˜k cos ϑ,v zz=1+˜k cos ϑ−2εi |k |˜k |q |1+˜kcos ϑ +m 2|q |2sin 2ϑ,v xy =εi +ε|q |ε,(12)where ˜k=|k |/ε,ϑis the lepton scattering angle,and m the mass of the emitted lepton,e.g.,the muon mass ≃105.9MeV.It was takenv T =1πIm Ψ0|J ν†(q )G (E f )J µ(q )|Ψ0 ,(15)in terms of the Green’s function G (E f )related to the nuclear Hamiltonian H ,i.e.,G (E f )=1E −E +iηχ(−)E (E )|.(18)The hadron tensor can be reduced to a single-particle expression and can be written in an expanded form as W µν(ω,q )=−1E f −εn −E +iη×T µνn (E ,E f −εn ),(19)where n denotes the eigenstate of the residual Hamil-tonian of A interacting nucleons related to the discreteeigenvalue εn andT µνn (E ,E )=λn ϕn |j ν†(q )1−V ′(E )j µ(q )|ϕn .(20)λn is the spectral strength of the hole state |ϕn ,which is the normalized overlap integral between |Ψ0 and |n ,while the factorπP∞Md E14The second matrix element in Eq.(20),withthe inclu-sionof√M+E +S †(E )−V †(E )σ·p Ψf+,(22)where S (E )and V (E )are the scalar and vector energy-dependent components of the relativistic optical po-tential for a nucleon with energy E [26].The upper component,Ψf+,is related to a two-component spinor,Φf ,which solves a Schr¨o dinger-like equation contain-ing equivalent central and spin-orbit potentials,obtained from the relativistic scalar and vector potentials [27,28],i.e.,Ψf+=M +E,(24)where D E (E )is the Darwin factor.As no relativistic optical potentials are available for the bound states,the wave functions ϕn are taken as the Dirac-Hartree solutions of a relativistic Lagrangian containing scalar and vector potentials [29,30].IV.RESULTSThe calculations have been performed with the same bound state wave functions and optical potentials as in Refs.[16,24,25],where the RDWIA was able to repro-duce (e,e ′p ),(γ,p ),and (e,e ′)data.The relativistic bound state wave functions have been obtained from Ref.[29],where relativistic Hartree-Bogoliubov equations are solved in the context of a rel-ativistic mean field theory and reproduce single-particle properties of several spherical and deformed nuclei [30].The scattering state is calculated by means of the energy-dependent and A-dependent EDAD1complex phenomenological optical potential of Ref.[26],which is fitted to proton elastic scattering data on several nuclei in an energy range up to 1040MeV.The initial states |ϕn are taken as single-particle one-hole states in the target.A pure shell model is assumed for the nuclear structure,i.e.,we take a unitary spectral strength for each single-particle state and the sum runs over all the occupied states.The results presented in the following contain the con-tribution of only the first term in Eq.(21).The cal-culation of the second term,which requires integration over all the eigenfunctions of the continuum spectrum of the optical potential,is a complicate task.Its contribu-tion has been estimated to be small in the kinematics explored;hence,it is neglected in the present calcula-tions.In order to show up the effect of the optical poten-tial on the inclusive reaction,the results obtained in the present approach are compared with those given by dif-ferent approximations.In the simplest approach the optical potential is ne-glected,i.e.,V =V †=0in Eq.(17),and the plane wave approximation is assumed for the final state wave functions χ(−)and ˜χ(−).In this plane wave impulse ap-proximation (PWIA)FSI between the outgoing nucleon and the residual nucleus are completely neglected.In another approach the imaginary part of the optical potential is neglected and only the real part is included.This approximation,that was sometimes used in the past,conserves the flux,but it is inconsistent with the exclu-sive process,where a complex optical potential must be used.Moreover,the use of a real optical potential is unsatisfactory from a theoretical point of view,since the optical potential has to be complex owing to the presence of open channels.The partial contribution given to the inclusive process by the sum of all the integrated exclusive reactions with one-nucleon emission is also shown in the following for aFIG.1:The cross sections of the16O(νµ,µ−)reaction,in-tegrated over the muon angle,for Eν=300,500,and1000 MeV.Solid lines represent the result of the Green’s function approach,dotted lines give PWIA,long-dashed lines show the result with a real optical potential,and dot-dashed lines the contribution of the integrated exclusive reactions with one-nucleon emission.Short dashed lines give the cross sections of the16O(¯νµ,µ+)reaction calculated with the Green’s func-tion approach.comparison.In this case only the eigenfunctionsχ(−)of V†are included and the imaginary part of the potential produces an absorption which does not conserve the total flux.We note that in the Green’s function approach of Eqs.(20)and(21)FSI are treated,consistently with the exclusive process,by means of a complex optical poten-tial.The imaginary part,however,does not produce a global loss offlux and it is responsible for the redistribu-tion of the strength among different channels.As afirst study case we have considered the16O tar-get nucleus,for which the adopted single-particle wave functions and optical potentials have given a good agree-ment between RDWIA calculations and(e,e′p),(γ,p), and(e,e′)data.In Fig.1the cross sections of the16O(νµ,µ−)reaction, integrated over the muon scattering angle,are displayed as a function of the muon kinetic energy Tµfor three dif-ferent values of the incident neutrino energy Eν=300,FIG.2:The cross sections of the16O(νµ,µ−)reaction,inte-grated over the muon energy,for Eν=300,500,and1000 MeV as a function of the scattering angle of the outgoing muonθµ.Line convention as in Fig.1.500and1000MeV.The behaviour of the calculated cross sections is similar for the different energies.The effect of the optical potential increases with Tµand decreases in-creasing Eν.At300MeV,the result of the PWIA is much higher than the one of the Green’s function ap-proach,while at1GeV the two results are almost the same but at the highest values of Tµ.The sum of the exclusive one-nucleon emission cross sections is always much smaller than the complete result.The difference indicates the relevance of inelastic channels and is due to the loss offlux produced by the absorptive imaginary part of the optical potential.In contrast,the cross sec-tions calculated with only the real part of the optical potential are practically the same as the ones obtained with the Green’s function approach.Although the use of a complex optical potential is conceptually important from a theoretical point of view,the negligible differences given by the two results mean that the conservation of flux,that is fulfilled in both calculations,is the most im-portant condition in the present situation.In contrast, significant differences are obtained with a real optical po-tential in the inclusive electron scattering[15,16].FIG.3:The total cross section of the 16O(νµ,µ−)reaction,integrated over the muon energy and angle,in terms of the neutrino energy E ν.Line convention as in Fig.1.Qualitatively similar results are obtained in Fig.2,where the cross sections integrated over the muon energy are displayed for E ν=300,500,and 1000MeV as a function of the scattering angle of the outgoing muon.The global effect of FSI is clearly shown in Fig.3,where the cross sections are integrated over the energy and the angle of the outgoing muon.The PWIA result is always much larger,while the loss of flux produced by the absorptive optical potential in the exclusive pro-cesses produces a too small cross section.In contrast,an optical potential with only a real component seems able to give a result comparable with the one of the Green’s function approach.Small differences are found only at higher neutrino energies.In Figs.1,2,and 3also the cross sections for the 16O(¯νµ,µ+)reaction are shown for a comparison.They are always much smaller than the corresponding cross sections with an incident neutrino.The different approaches have been compared with the experimental results of the LSND collaboration at Los Alamos for the 12C(νµ,µ−)reaction [1,2,3].The cal-culations have been flux-averaged over the Los Alamos neutrino spectrum.In Fig.4the Green’s function ap-proach is normalized to the experimental data,while the other results are accordingly scaled.The shape of experi-mental data is reasonably reproduced.The flux-averagedFIG.4:The distribution of the muon kinetic energy for the in-clusive 12C(νµ,µ−)reaction.Experimental data from Ref.[2].The result of the Green’s function approach is normalized to the experimental data.The other curves are scaled,accord-ingly.Line convention as in Fig.1.cross section integrated over the muon energy gives 11.15×10−40cm 2.At the low energy of the experiment,how-ever,other processes,which are here not included,be-sides the quasielastic scattering can affect the inclusive reaction,in particular the excitation of the discrete states of 12C.The experimental value (10.6±0.3±1.8)×10−40cm 2[3]is slightly overestimated.The results obtained by other calculations [4,6,7,8,10,11,13]give larger values.V.SUMMARY AND CONCLUSIONSWe have applied to (νµ,µ−)and (¯νµ,µ+)reactions an approach based on the spectralization of the single-particle Green’s function in terms of the eigenfunctions of the complex optical potential and of its Hermitian con-jugate.This approach has proved to be rather successful in describing inclusive electron scattering.Its advantage stands in the fact that it is able to include in a simple way and keeping flux conservation the final state inter-actions by using an optical potential which is essential to reproduce exclusive electron knockout reactions.The method is applied within a relativistic framework to weak charged-current reactions for an energy up to 1GeV,where nuclear structure effects are important and dominant with respect to nucleon-resonance excitations. The reaction mechanism is assumed to be a direct one, where the incident neutrino interacts with only one neu-tron and a proton is emitted together with a negative-charged muon.A single-particle model is used to de-scribe the structure of the nucleus and a sum over all single-particle occupied states is performed. Calculations for the16O target nucleus have been pre-sented at neutrino energies up to1GeV.The optical potential andflux conservation have a large effect on the cross sections.For12C,the results averaged over the ex-perimentalflux of neutrinos are compared with the avail-able data.A fair agreement is obtained in arbitrary units. 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