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128-600: Malvin Ruderman and Charles Kittel of the University of California, Berkeley first proposed the model to explain unusually broad nuclear spin resonance lines in natural metallic silver. The theory is an indirect exchange coupling : the hyperfine interaction couples the nuclear spin of one atom to a conduction electron also coupled to the spin of a different nucleus. The assumption of hyperfine interaction turns out to be unnecessary, and can be replaced equally well with

256-1414: A potential energy surface (PES) V ( R 1 , R 2 , … , R N ) {\displaystyle V(\mathbf {R} _{1},\mathbf {R} _{2},\ldots ,\mathbf {R} _{N})} . It is assumed that the functional dependence of V on its coordinates is such that V ( R 1 , R 2 , … , R N ) = V ( R 1 ′ , R 2 ′ , … , R N ′ ) {\displaystyle V(\mathbf {R} _{1},\mathbf {R} _{2},\ldots ,\mathbf {R} _{N})=V(\mathbf {R} '_{1},\mathbf {R} '_{2},\ldots ,\mathbf {R} '_{N})} for R i ′ = R i + t (translation) and R i ′ = R i + Δ ϕ | s | ( s × R i ) (infinitesimal rotation) , {\displaystyle \mathbf {R} '_{i}=\mathbf {R} _{i}+\mathbf {t} \;\;{\text{(translation) and}}\;\;\mathbf {R} '_{i}=\mathbf {R} _{i}+{\frac {\Delta \phi }{|\mathbf {s} |}}\;(\mathbf {s} \times \mathbf {R} _{i})\;\;{\text{(infinitesimal rotation)}},} where t and s are arbitrary vectors and Δφ

384-442: A " Public Ivy ". Since its inaugural 1990 reputational survey, Times Higher Education has considered Berkeley to be one of the world's "six super brands" along with the University of Oxford and the University of Cambridge , Harvard University , MIT , and Stanford University . The 2010 United States National Research Council Rankings identified Berkeley as having the highest number of top-ranked doctoral programs in

512-1263: A Taylor expansion of V around the equilibrium geometry, V = V 0 + ∑ i = 1 N ∑ α = 1 3 ( ∂ V ∂ ρ i α ) 0 ρ i α + 1 2 ∑ i , j = 1 N ∑ α , β = 1 3 ( ∂ 2 V ∂ ρ i α ∂ ρ j β ) 0 ρ i α ρ j β + ⋯ , {\displaystyle V=V_{0}+\sum _{i=1}^{N}\sum _{\alpha =1}^{3}{\Big (}{\frac {\partial V}{\partial \rho _{i\alpha }}}{\Big )}_{0}\;\rho _{i\alpha }+{\frac {1}{2}}\sum _{i,j=1}^{N}\sum _{\alpha ,\beta =1}^{3}{\Big (}{\frac {\partial ^{2}V}{\partial \rho _{i\alpha }\partial \rho _{j\beta }}}{\Big )}_{0}\;\rho _{i\alpha }\rho _{j\beta }+\cdots ,} and truncate after three terms (the so-called harmonic approximation), we can describe V with only

640-478: A blue background with an imaginary yellow pen. The California Victory Cannon, placed on Tightwad Hill overlooking the stadium, is fired before every football home game, after every score, and after every Cal victory. First used in the 1963 Big Game, it was originally placed on the sidelines before moving to Tightwad Hill in 1971. The only time the cannon ran out of ammunition was during a game against Pacific in 1991, when Cal scored 12 touchdowns. The Cal Mic Men,

768-548: A culvert upstream of the recreational complex at the mouth of Strawberry Canyon and passes beneath California Memorial Stadium before appearing again in Faculty Glade. It then runs through the center of the campus before disappearing underground at the west end of campus. The north fork appears just east of University House and runs through the glade north of the Valley Life Sciences Building,

896-536: A difference vector, distance is invariant under translation of the frame as well. By quantizing the classical energy in Hamilton form one obtains the a molecular Hamilton operator that is often referred to as the Coulomb Hamiltonian . This Hamiltonian is a sum of five terms. They are Here M i is the mass of nucleus i , Z i is the atomic number of nucleus i , and m e is the mass of

1024-424: A frame that moves with the nuclei, so that the nuclei are at rest with respect to this frame. The frame stays parallel to a space-fixed frame. It is an inertial frame because the nuclei are assumed not to be accelerated by external forces or torques. The origin of the frame is arbitrary, it is usually positioned on a central nucleus or in the nuclear center of mass. Sometimes it is stated that the nuclei are "at rest in

1152-507: A function of the distance between the layers. This ferromagnetic/antiferromagnetic oscillation is one prediction of the RKKY theory. University of California, Berkeley The University of California, Berkeley ( UC Berkeley , Berkeley , Cal , or California ) is a public land-grant research university in Berkeley, California , United States. Founded in 1868 and named after

1280-501: A kinetic energy operator. The second term is known as the mass polarization term. The translationally invariant Hamiltonian H ′ {\displaystyle H'} can be shown to be self-adjoint and to be bounded from below. That is, its lowest eigenvalue is real and finite. Although H ′ {\displaystyle H'} is necessarily invariant under permutations of identical particles (since H {\displaystyle H} and

1408-431: A molecule are the nuclei, characterized by their atomic numbers , Z , and the electrons, which have negative elementary charge , − e . Their interaction gives a nuclear charge of Z  +  q , where q = − eN , with N equal to the number of electrons. Electrons and nuclei are, to a very good approximation, point charges and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are

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1536-581: A new center for computing and data science. For the 2020 fiscal year, Berkeley set a fundraising record, receiving over $ 1 billion in gifts and pledges, and two years later, it broke that record, raising over $ 1.2 billion. Officially named the "University of California, Berkeley" it is often shortened to "Berkeley" in general reference or in an academic context ( Berkeley Law , Berkeley Engineering , Berkeley Haas , Berkeley Public Health ) and to "California" or "Cal" particularly when referring to its athletic teams ( California Golden Bears ). In August 2022,

1664-488: A parity violating term predicted by the Standard Model must be mentioned. Although it is an extremely small interaction, it has attracted a fair amount of attention in the scientific literature because it gives different energies for the enantiomers in chiral molecules . The remaining part of this article will ignore spin terms and consider the solution of the eigenvalue (time-independent Schrödinger) equation of

1792-548: A point particle with mass equal to the sum M tot of the masses of all the particles. In quantum mechanics a free particle has as state function a plane wave function, which is a non-square-integrable function of well-defined momentum. The kinetic energy of this particle can take any positive value. The position of the COM is uniformly probable everywhere, in agreement with the Heisenberg uncertainty principle . By introducing

1920-435: A print and online edition. Berkeley's FM student radio station , KALX , broadcasts on 90.7 MHz. It is run largely by volunteers, including both students and community members. Berkeley also features an assortment of student-run publications: There are ninety-four political student groups on campus, including MEChXA de UC Berkeley, Berkeley ACLU , Berkeley Students for Life, Campus Greens, The Sustainability Team (STEAM),

2048-702: A research facility and headquarters for the California Institute for Quantitative Biosciences , opened. Supported by a grant from alumnus James Simons , the Simons Institute for the Theory of Computing was established in 2012. In 2015, Berkeley and its sister campus, UCSF , established the Innovative Genomics Institute to develop CRISPR gene editing , and, in 2020, an anonymous donor pledged $ 252 million to help fund

2176-1259: A separation we must distinguish internal and external coordinates, to which end Eckart introduced conditions to be satisfied by the coordinates. We will show how these conditions arise in a natural way from a harmonic analysis in mass-weighted Cartesian coordinates. In order to simplify the expression for the kinetic energy we introduce mass-weighted displacement coordinates ρ i ≡ M i ( R i − R i 0 ) {\displaystyle {\boldsymbol {\rho }}_{i}\equiv {\sqrt {M_{i}}}(\mathbf {R} _{i}-\mathbf {R} _{i}^{0})} . Since ∂ ∂ ρ i α = ∂ M i ( ∂ R i α − ∂ R i α 0 ) = 1 M i ∂ ∂ R i α , {\displaystyle {\frac {\partial }{\partial \rho _{i\alpha }}}={\frac {\partial }{{\sqrt {M_{i}}}(\partial R_{i\alpha }-\partial R_{i\alpha }^{0})}}={\frac {1}{\sqrt {M_{i}}}}{\frac {\partial }{\partial R_{i\alpha }}},}

2304-487: A space-fixed frame". This statement implies that the nuclei are viewed as classical particles, because a quantum mechanical particle cannot be at rest. (It would mean that it had simultaneously zero momentum and well-defined position, which contradicts Heisenberg's uncertainty principle). Since the nuclear positions are constants, the electronic kinetic energy operator is invariant under translation over any nuclear vector. The Coulomb potential, depending on difference vectors,

2432-538: A standard at home football games, has recently expanded to involve basketball and volleyball. The traditional role comes from students holding megaphones and yelling, but now includes microphones, a dedicated platform during games, and the direction of the entire student section. Berkeley students are offered a variety of housing options, including university-owned or affiliated residences, private residences, fraternities and sororities, and cooperative housing (co-ops). Berkeley students, and those of other local schools, have

2560-532: A student regent, and a non-voting student regent-designate. Prior to 1952, Berkeley was the University of California, so the university president was also Berkeley's chief executive. In 1952, the university reorganized itself into a system of semi-autonomous campuses, with each campus having its own chief executive, a chancellor, who would, in turn, report to the president of the university system. Twelve vice-chancellors report directly to Berkeley's chancellor, and

2688-612: A system of semi-autonomous campuses, with each campus given a chancellor, and Clark Kerr became Berkeley's first Chancellor, while   Robert Sproul remained in place as the President of the University of California. Berkeley gained a worldwide reputation for political activism in the 1960s. In 1964, the Free Speech Movement organized student resistance to the university's restrictions on political activities on campus—most conspicuously, student activities related to

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2816-543: A trustee of the College of California, suggested that a new campus site north of Oakland be named in honor of Anglo-Irish philosopher George Berkeley . The university began admitting women the following year. In 1870, Henry Durant , founder of the College of California, became its first president. With the completion of North and South Halls in 1873, the university relocated to its Berkeley location with 167 male and 22 female students. The first female student to graduate

2944-458: A university task force was formed which recommended renaming the athletic identity to "Cal Berkeley" to further tie the athletic brand to academic prestige, and reduce public confusion. The University of California is governed by a twenty-six member Board of Regents , eighteen of whom are appointed by the Governor of California to 12-year terms. The board also has seven ex officio members,

3072-584: Is Oski the Bear , who debuted in 1941. Previously, live bear cubs were used as mascots at Memorial Stadium until it was decided in 1940 that a costumed mascot would be a better alternative. Named after the Oski-wow-wow yell, he is cared for by the Oski Committee, whose members have exclusive knowledge of the identity of the costume-wearer. The University of California Marching Band , which has served

3200-493: Is classified among "R1: Doctoral Universities – Very high research activity". The Lawrence Berkeley National Laboratory was originally founded as part of the university. Berkeley was a founding member of the Association of American Universities and was one of the original eight " Public Ivy " schools. In 2021, the federal funding for campus research and development exceeded $ 1 billion. Thirty-two libraries also compose

3328-577: Is a component of an operator known as the vibrational angular momentum operator (although it does not satisfy angular momentum commutation relations), Π α = − i ℏ ∑ s , t = 1 3 N − 6 ζ s t α q s ∂ ∂ q t {\displaystyle \Pi _{\alpha }=-i\hbar \sum _{s,t=1}^{3N-6}\zeta _{st}^{\alpha }\;q_{s}{\frac {\partial }{\partial q_{t}}}} with

3456-661: Is also home to the University of California Botanical Garden , with more than 12,000 individual species. What is considered the historic campus today was the result of the 1898 "International Competition for the Phoebe Hearst Architectural Plan for the University of California," funded by William Randolph Hearst 's mother and initially held in the Belgian city of Antwerp ; eleven finalists were judged again in San Francisco in 1899. The winner

3584-718: Is also often called the electronic Hamiltonian, describes the energy of the electrons in the electrostatic field of the nuclei, where the nuclei are assumed to be stationary with respect to an inertial frame. The form of the electronic Hamiltonian is H ^ e l = T ^ e + U ^ e n + U ^ e e + U ^ n n . {\displaystyle {\hat {H}}_{\mathrm {el} }={\hat {T}}_{e}+{\hat {U}}_{en}+{\hat {U}}_{ee}+{\hat {U}}_{nn}.} The coordinates of electrons and nuclei are expressed with respect to

3712-490: Is an important symbol of California school spirit. The Big "C" has its roots in an early 20th-century campus event called "Rush," which pitted the freshman and sophomore classes against each other in a race up Charter Hill that often developed into a wrestling match. It was eventually decided to discontinue Rush and, in 1905, the freshman and sophomore classes banded together in a show of unity to build "the Big C." Students invented

3840-503: Is an infinitesimal angle, Δφ >> Δφ . This invariance condition on the PES is automatically fulfilled when the PES is expressed in terms of differences of, and angles between, the R i , which is usually the case. In the remaining part of this article we assume that the molecule is semi-rigid . In the second step of the BO approximation the nuclear kinetic energy T n is reintroduced and

3968-526: Is common to follow Podolsky by writing down the Laplace–Beltrami operator in the same (generalized, curvilinear) coordinates s as used for the classical form. The equation for this operator requires the inverse of the metric tensor g and its determinant. Multiplication of the Laplace–Beltrami operator by − ℏ 2 {\displaystyle -\hbar ^{2}} gives

RKKY interaction - Misplaced Pages Continue

4096-400: Is considered. One would like to recognize in its solution: the motion of the nuclear center of mass (3 degrees of freedom), the overall rotation of the molecule (3 degrees of freedom), and the nuclear vibrations. In general, this is not possible with the given nuclear kinetic energy, because it does not separate explicitly the 6 external degrees of freedom (overall translation and rotation) from

4224-620: Is decentralized; applicants apply directly to the department or degree program. Most graduate students are supported by fellowships, teaching assistantships, or research assistantships. Doe Library serves as the Berkeley library system 's reference, periodical, and administrative center, while most of the main collections reside in the subterranean Gardner Main Stacks and Moffitt Undergraduate Library. The Bancroft Library , which has over 400,000 printed volumes and 70 million manuscripts, pictures, and maps, maintains special collections that document

4352-475: Is easily solved, it factorizes into 3 N  − 6 equations for one-dimensional harmonic oscillators . The main effort in this approximate solution of the nuclear motion Schrödinger equation is the computation of the Hessian F of V and its diagonalization. This approximation to the nuclear motion problem, described in 3 N mass-weighted Cartesian coordinates, became standard in quantum chemistry , since

4480-484: Is generally assumed that the solution of the time-independent Schrödinger equation associated with the Coulomb Hamiltonian will predict most properties of the molecule, including its shape (three-dimensional structure), calculations based on the full Coulomb Hamiltonian are very rare. The main reason is that its Schrödinger equation is very difficult to solve. Applications are restricted to small systems like

4608-579: Is invariant as well. In the description of atomic orbitals and the computation of integrals over atomic orbitals this invariance is used by equipping all atoms in the molecule with their own localized frames parallel to the space-fixed frame. As explained in the article on the Born–Oppenheimer approximation , a sufficient number of solutions of the Schrödinger equation of H el {\displaystyle H_{\text{el}}} leads to

4736-513: Is often referred to as Watson's nuclear motion Hamiltonian , but it is also known as the Eckart Hamiltonian . The algebraic form of many observables—i.e., Hermitian operators representing observable quantities—is obtained by the following quantization rules : Classically the electrons and nuclei in a molecule have kinetic energy of the form p /(2  m ) and interact via Coulomb interactions , which are inversely proportional to

4864-536: Is one of the few systems that allows an exact solution of the Schrödinger equation. Alternatively, the nuclear motion (rovibrational) Schrödinger equation can be solved in a special frame (an Eckart frame ) that rotates and translates with the molecule. Formulated with respect to this body-fixed frame the Hamiltonian accounts for rotation , translation and vibration of the nuclei. Since Watson introduced in 1968 an important simplification to this Hamiltonian, it

4992-498: Is the Levi-Civita symbol . The terms quadratic in the P α {\displaystyle {\mathcal {P}}_{\alpha }} are centrifugal terms, those bilinear in P α {\displaystyle {\mathcal {P}}_{\alpha }} and Π β {\displaystyle \Pi _{\beta }\,} are Coriolis terms. The quantities Q s, iγ are

5120-533: Is the center of mass term X ≡ 1 M t o t ∑ i = 1 N M i R i w i t h M t o t ≡ ∑ i = 1 N M i . {\displaystyle \mathbf {X} \equiv {\frac {1}{M_{\mathrm {tot} }}}\sum _{i=1}^{N}M_{i}\mathbf {R} _{i}\quad \mathrm {with} \quad M_{\mathrm {tot} }\equiv \sum _{i=1}^{N}M_{i}.} The second term

5248-467: Is the kinetic energy of the COM motion, which can be treated separately since H ′ {\displaystyle H'} does not depend on X . As just stated, its eigenstates are plane waves. The potential V ( t ) consists of the Coulomb terms expressed in the new coordinates. The first term of H ′ {\displaystyle H'} has the usual appearance of

RKKY interaction - Misplaced Pages Continue

5376-414: Is the rotational term akin to the kinetic energy of the rigid rotor . Here P α {\displaystyle {\mathcal {P}}_{\alpha }} is the α component of the body-fixed rigid rotor angular momentum operator , see this article for its expression in terms of Euler angles . The operator Π α {\displaystyle \Pi _{\alpha }\,}

5504-583: The Anglo-Irish philosopher George Berkeley , it is the state's first land-grant university and is the founding campus of the University of California system. Berkeley has an enrollment of more than 45,000 students. The university is organized around fifteen schools of study on the same campus, including the College of Chemistry , the College of Engineering , and the Haas School of Business . It

5632-580: The Berkeley Student Food Collective , Students for Sensible Drug Policy, Cal Berkeley Democrats, and the Berkeley College Republicans. The Residence Hall Assembly (RHA) is the student-led umbrella organization that oversees event planning, legislation, sponsorships and other activities for over 7,200 on-campus undergraduate residents. Berkeley students also run a number of consulting groups, including

5760-808: The Berkeley library system which is the sixth largest research library by number of volumes held in the United States. Berkeley students compete in thirty varsity athletic sports, and the university is one of eighteen full-member institutions in the Atlantic Coast Conference (ACC). Berkeley's athletic teams, the California Golden Bears , have also won 107 national championships, 196 individual national titles, and 223 Olympic medals (including 121 gold). Berkeley's alumni , faculty, and researchers include 55 Nobel laureates and 19 Academy Award winners, and

5888-673: The Civil Rights Movement . The arrest in Sproul Plaza of Jack Weinberg , a recent Berkeley alumnus and chair of Campus CORE , prompted a series of student-led acts of formal remonstrance and civil disobedience that ultimately gave rise to the Free Speech Movement, which movement would prevail and serve as a precedent for student opposition to America's involvement in the Vietnam War. In 1982,

6016-688: The Coriolis coupling constant : ζ s t α = ∑ i = 1 N ∑ β , γ = 1 3 ϵ α β γ Q s , i β Q t , i γ a n d α = 1 , 2 , 3. {\displaystyle \zeta _{st}^{\alpha }=\sum _{i=1}^{N}\sum _{\beta ,\gamma =1}^{3}\epsilon _{\alpha \beta \gamma }Q_{s,i\beta }\,Q_{t,i\gamma }\;\;\mathrm {and} \quad \alpha =1,2,3.} Here ε αβγ

6144-606: The Hamiltonian , R ij is the distance between the nuclei i and j , I i is the nuclear spin of atom i , Δ k m k m is a matrix element that represents the strength of the hyperfine interaction, m is the effective mass of the electrons in the crystal, and k m is the Fermi momentum . Intuitively, we may picture this as when one magnetic atom scatters an electron wave, which then scatters off another magnetic atom many atoms away, thus coupling

6272-701: The Lawrence Hall of Science . The Museum of Paleontology, found in the lobby of the Valley Life Sciences Building, showcases a variety of dinosaur fossils including a complete cast of a Tyrannosaurus Rex. The campus also offers resources for innovation and entrepreneurship, such as the Big Ideas Competition, the Sutardja Center for Entrepreneurship and Technology, and the Berkeley Haas Innovation Lab. The campus

6400-656: The Mathematical Sciences Research Institute (MSRI) was established on campus with support from the National Science Foundation and at the request of three Berkeley mathematicians— Shiing-Shen Chern , Calvin Moore , and Isadore M. Singer . The institute is now widely regarded as a leading center for collaborative mathematical research, drawing thousands of visiting researchers from around the world each year. In

6528-612: The National Register of Historic Places . Built in 1873 in a Victorian Second-Empire-style , South Hall, designed by David Farquharson, is the oldest university building in California. It, and the Frederick Law Olmsted -designed Piedmont Avenue east of the main campus, are two of the only surviving examples of the nineteenth-century campus. Other notable architects and firms whose work can be found in

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6656-619: The Sather Tower (nicknamed "the Campanile" after its architectural inspiration, St Mark's Campanile in Venice), the tallest university clock tower in the United States. Buildings he regarded as temporary and non-academic were designed in shingle or Collegiate Gothic styles; examples of these are North Gate Hall, Dwinelle Annex, and Stephens Hall. Many of Howard's designs are recognized California Historical Landmarks and are listed on

6784-774: The Space Sciences Laboratory , the Mathematical Sciences Research Institute , an 800-acre (320-hectare) ecological preserve, the University of California Botanical Garden and a recreation center in Strawberry Canyon. Portions of the mostly undeveloped, eastern area of the campus are actually within the City of Oakland ; these portions extend from the Claremont Resort north through the Panoramic Hill neighborhood to Tilden Park . To

6912-797: The University of California, Los Angeles . In 1905, the University Farm was established near Sacramento , ultimately becoming the University of California, Davis . In 1919, the Los Angeles branch of the California State Normal School became the southern branch of the university, which ultimately became the University of California, Los Angeles . By the 1920s, the number of campus buildings in Berkeley had grown substantially and included twenty structures designed by architect John Galen Howard . In 1917, one of

7040-1112: The chain rule one can show that H = − ℏ 2 2 M tot ∇ X 2 + H ′ with  H ′ = − ℏ 2 2 ∑ i = 1 N tot − 1 1 m i ∇ i 2 + ℏ 2 2 M tot ∑ i , j = 1 N tot − 1 ∇ i ⋅ ∇ j + V ( t ) . {\displaystyle H=-{\frac {\hbar ^{2}}{2M_{\textrm {tot}}}}\nabla _{\mathbf {X} }^{2}+H'\quad {\text{with }}\quad H'=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N_{\textrm {tot}}-1}{\frac {1}{m_{i}}}\nabla _{i}^{2}+{\frac {\hbar ^{2}}{2M_{\textrm {tot}}}}\sum _{i,j=1}^{N_{\textrm {tot}}-1}\nabla _{i}\cdot \nabla _{j}+V(\mathbf {t} ).} The first term of H {\displaystyle H}

7168-1305: The chain rule . The Watson Hamiltonian, describing all motions of the N nuclei, is H ^ = − ℏ 2 2 M t o t ∑ α = 1 3 ∂ 2 ∂ X α 2 + 1 2 ∑ α , β = 1 3 μ α β ( P α − Π α ) ( P β − Π β ) + U − ℏ 2 2 ∑ s = 1 3 N − 6 ∂ 2 ∂ q s 2 + V . {\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2M_{\mathrm {tot} }}}\sum _{\alpha =1}^{3}{\frac {\partial ^{2}}{\partial X_{\alpha }^{2}}}+{\frac {1}{2}}\sum _{\alpha ,\beta =1}^{3}\mu _{\alpha \beta }({\mathcal {P}}_{\alpha }-\Pi _{\alpha })({\mathcal {P}}_{\beta }-\Pi _{\beta })+U-{\frac {\hbar ^{2}}{2}}\sum _{s=1}^{3N-6}{\frac {\partial ^{2}}{\partial q_{s}^{2}}}+V.} The first term

7296-419: The curvilinear coordinates s = ( s i ) through 2 T = ∑ i j g i j s ˙ i s ˙ j . {\displaystyle 2T=\sum _{ij}g_{ij}{\dot {s}}_{i}{\dot {s}}_{j}.} The quantization step is the transformation of this classical kinetic energy into a quantum mechanical operator. It

7424-853: The distance r ij between particle i and j . r i j ≡ | r i − r j | = ( r i − r j ) ⋅ ( r i − r j ) = ( x i − x j ) 2 + ( y i − y j ) 2 + ( z i − z j ) 2 . {\displaystyle r_{ij}\equiv |\mathbf {r} _{i}-\mathbf {r} _{j}|={\sqrt {(\mathbf {r} _{i}-\mathbf {r} _{j})\cdot (\mathbf {r} _{i}-\mathbf {r} _{j})}}={\sqrt {(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}}}.} In this expression r i stands for

7552-1112: The exchange interaction . The simplest treatment assumes a Bloch wavefunction basis and therefore only applies to crystalline systems; the resulting correlation energy, computed with perturbation theory , takes the following form: H ( R i j ) = I i ⋅ I j 4 | Δ k m k m | 2 m ∗ ( 2 π ) 3 R i j 4 ℏ 2 [ 2 k m R i j cos ⁡ ( 2 k m R i j ) − sin ⁡ ( 2 k m R i j ) ] , {\displaystyle H(\mathbf {R} _{ij})={\frac {\mathbf {I} _{i}\cdot \mathbf {I} _{j}}{4}}{\frac {\left|\Delta _{k_{m}k_{m}}\right|^{2}m^{*}}{(2\pi )^{3}R_{ij}^{4}\hbar ^{2}}}\left[2k_{m}R_{ij}\cos(2k_{m}R_{ij})-\sin(2k_{m}R_{ij})\right]{\text{,}}} where H represents

7680-463: The kinetic energies of the electrons and the Coulomb (electrostatic) interactions between the two kinds of charged particles. The Hamiltonian that contains only the kinetic energies of electrons and nuclei, and the Coulomb interactions between them, is known as the Coulomb Hamiltonian . From it are missing a number of small terms, most of which are due to electronic and nuclear spin . Although it

7808-478: The molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule . This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity , specific heat , electrical conductivity , optical , and magnetic properties , and reactivity . The elementary parts of

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7936-484: The "University shall have for its design, to provide instruction and thorough and complete education in all departments of science, literature and art, industrial and professional pursuits, and general education, and also special courses of instruction in preparation for the professions." Ten faculty members and forty male students made up the fledgling university when it opened in Oakland in 1869. Frederick Billings ,

8064-506: The (internal) atomic vibrations enter the problem. Further, for molecules larger than triatomic ones, it is quite common to introduce the harmonic approximation , which approximates the potential energy surface as a quadratic function of the atomic displacements. This gives the harmonic nuclear motion Hamiltonian . Making the harmonic approximation, we can convert the Hamiltonian into a sum of uncoupled one-dimensional harmonic oscillator Hamiltonians. The one-dimensional harmonic oscillator

8192-649: The 1930s, Ernest Orlando Lawrence helped establish the Radiation Laboratory (now Lawrence Berkeley National Laboratory ) and invented the cyclotron , which won him the Nobel physics prize in 1939. Using the cyclotron, Berkeley professors and Berkeley Lab researchers went on to discover sixteen chemical elements —more than any other university in the world. In particular, during World War II and following Glenn Seaborg 's then-secret discovery of plutonium, Lawrence's Radiation Laboratory began to contract with

8320-462: The 3 N  − 6 internal degrees of freedom. In fact, the kinetic energy operator here is defined with respect to a space-fixed (SF) frame. If we were to move the origin of the SF frame to the nuclear center of mass, then, by application of the chain rule , nuclear mass polarization terms would appear. It is customary to ignore these terms altogether and we will follow this custom. In order to achieve

8448-571: The Berkeley Group, founded in 2003 and affiliated with the Haas School. Students from various concentrations are recruited and trained to work on pro-bono consulting engagements with actual nonprofit clients. Berkeley Consulting, founded in 1996, has served over 140 companies across the high-tech, retail, banking, and non-profit sectors. Molecular Hamiltonian In atomic, molecular, and optical physics and quantum chemistry ,

8576-557: The Berkeley campus is in the city limits of Berkeley with portion of the property extending into Oakland . It encompasses approximately 1,232-acres, though the "central campus" occupies only the low-lying western 178-acres of this area. Of the remaining acres, approximately 200-acres are occupied by the Lawrence Berkeley National Laboratory; other facilities above the main campus include the Lawrence Hall of Science and several research units, notably

8704-519: The Bill and Melinda Gates Foundation, billionaire Sir Li Ka-Shing , Israeli-Russian billionaire Yuri Milner , Thomas and Stacey Siebel , Sanford and Joan Weill , and professor Gordon Rausser ($ 50 million gift in 2020). Hundreds of millions of dollars have been given anonymously. The 2008–13 "Campaign for Berkeley" raised $ 3.13 billion from 281,855 donors, and the "Light the Way" campaign, which concluded at

8832-414: The COM kinetic energy are invariant), its invariance is not manifest. Not many actual molecular applications of H ′ {\displaystyle H'} exist; see, however, the seminal work on the hydrogen molecule for an early application. In the great majority of computations of molecular wavefunctions the electronic problem is solved with the clamped nucleus Hamiltonian arising in

8960-403: The Coulomb Hamiltonian. The Coulomb Hamiltonian has a continuous spectrum due to the center of mass (COM) motion of the molecule in homogeneous space. In classical mechanics it is easy to separate off the COM motion of a system of point masses. Classically the motion of the COM is uncoupled from the other motions. The COM moves uniformly (i.e., with constant velocity) through space as if it were

9088-587: The Haas ( Walter A. Haas , Rhoda Haas Goldman , Walter A. Haas Jr. , Peter E. Haas , Bob Haas ) family. Berkeley has also benefited from benefactors beyond its alumni ranks, notable among which are Mark Zuckerberg and Priscilla Chan ; Vitalik Buterin , Patrick Collison , John Collison , the Ron Conway family, Daniel Gross , Dustin Moskovitz and Cari Tuna , along with Jane Street principals; BP ;

9216-586: The Haas School of Business (1), College of Chemistry (5), College of Engineering (20), College of Environmental Design (4), College of Letters and Science (67), Rausser College of Natural Resources (10), and individual majors (2). The most popular majors are electrical engineering and computer sciences , political science , molecular and cell biology , environmental science , and economics . Requirements for undergraduate degrees include an entry-level writing requirement before enrollment (typically fulfilled by minimum scores on standardized admissions exams such as

9344-532: The SAT or ACT), completing coursework on "American History and Institutions" before or after enrollment by taking an introductory class, passing an "American Cultures Breadth" class at Berkeley, as well as requirements for reading and composition and specific requirements declared by the department and school. Berkeley has a "comprehensive" graduate program, with high coexistence with the programs offered to undergraduates, and offers interdisciplinary graduate programs with

9472-694: The Schrödinger equation with Hamiltonian H ^ n u c = − ℏ 2 2 ∑ i = 1 N ∑ α = 1 3 1 M i ∂ 2 ∂ R i α 2 + V ( R 1 , … , R N ) {\displaystyle {\hat {H}}_{\mathrm {nuc} }=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}\sum _{\alpha =1}^{3}{\frac {1}{M_{i}}}{\frac {\partial ^{2}}{\partial R_{i\alpha }^{2}}}+V(\mathbf {R} _{1},\ldots ,\mathbf {R} _{N})}

9600-494: The U.S. Army to develop the atomic bomb. Physics professor J. Robert Oppenheimer was named scientific head of the Manhattan Project in 1942. Along with the Lawrence Berkeley National Laboratory, Berkeley founded and was then a partner in managing two other labs, Los Alamos National Laboratory (1943) and Lawrence Livermore National Laboratory (1952). In 1952, the University of California reorganized itself into

9728-458: The UC Office of the President and accounts for 12 percent of Berkeley's total revenues. Berkeley has benefited from private philanthropy and alumni and their foundations have given to the university for operations and capital expenditures with the more prominent being J. Paul Getty , Ann Getty , Sanford Diller , Donald Fisher , Flora Lamson Hewlett , David Schwartz ( Bio-Rad ) and members of

9856-700: The University of California (ASUC) is the official student association that controls funding for student groups and organizes on-campus student events. The two main political parties are "Student Action" and "CalSERVE." The organization was founded in 1887 and has an annual operating budget of $ 1.7 million (excluding the budget of the Graduate Assembly of the ASUC), in addition to various investment assets. Its alumni include multiple State Senators, Assemblymembers, and White House Administration officials. Berkeley's student-run online television station, CalTV ,

9984-459: The campus and surrounding area are Bernard Maybeck (Faculty Club); Julia Morgan (Hearst Women's Gymnasium and Julia Morgan Hall ); William Wurster (Stern Hall); Moore Ruble Yudell (Haas School of Business); Tod Williams Billie Tsien Architects (C.V. Starr East Asian Library), and Diller Scofidio + Renfro (Berkeley Art Museum and Pacific Film Archive). Flowing into the main campus are two branches of Strawberry Creek . The south fork enters

10112-576: The college football tradition of card stunts . Then known as Bleacher Stunts, they were first performed during the 1910 Big Game and consisted of two stunts: a picture of the Stanford Axe and a large blue "C" on a white background. The tradition is continued today by the Rally Committee in the Cal student section and incorporates complicated motions, for example tracing the Cal script logo on

10240-596: The components of the normal coordinates introduced above. Alternatively, normal coordinates may be obtained by application of Wilson's GF method . The 3 × 3 symmetric matrix μ {\displaystyle {\boldsymbol {\mu }}} is called the effective reciprocal inertia tensor . If all q s were zero (rigid molecule) the Eckart frame would coincide with a principal axes frame (see rigid rotor ) and μ {\displaystyle {\boldsymbol {\mu }}} would be diagonal, with

10368-416: The coordinate vector X of the center of mass as three of the degrees of freedom of the system and eliminating the coordinate vector of one (arbitrary) particle, so that the number of degrees of freedom stays the same, one obtains by a linear transformation a new set of coordinates t i . These coordinates are linear combinations of the old coordinates of all particles (nuclei and electrons). By applying

10496-410: The coordinate vector of any particle (electron or nucleus), but from here on we will reserve capital R to represent the nuclear coordinate, and lower case r for the electrons of the system. The coordinates can be taken to be expressed with respect to any Cartesian frame centered anywhere in space, because distance, being an inner product, is invariant under rotation of the frame and, being the norm of

10624-490: The current century, Berkeley has become less politically active, although more liberal. Democrats outnumber Republicans on the faculty by a ratio of nine to one, which is a ratio similar to that of American academia generally. The school has become more focused on STEM disciplines and fundraising. In 2007, the Energy Biosciences Institute was established with funding from BP and Stanley Hall,

10752-464: The days (1980s-1990s) that algorithms for accurate computations of the Hessian F became available. Apart from the harmonic approximation, it has as a further deficiency that the external (rotational and translational) motions of the molecule are not accounted for. They are accounted for in a rovibrational Hamiltonian that sometimes is called Watson's Hamiltonian . In order to obtain a Hamiltonian for external (translation and rotation) motions coupled to

10880-400: The deans of the fifteen colleges and schools report to the executive vice chancellor and provost, Berkeley's chief academic officer. Twenty-three presidents and chancellors have led Berkeley since its founding. Presidents Chancellors With the exception of government contracts, public support is apportioned to Berkeley and the other campuses of the University of California system through

11008-782: The electron. The Laplace operator of particle i is: ∇ r i 2 ≡ ∇ r i ⋅ ∇ r i = ∂ 2 ∂ x i 2 + ∂ 2 ∂ y i 2 + ∂ 2 ∂ z i 2 {\displaystyle \nabla _{\mathbf {r} _{i}}^{2}\equiv {\boldsymbol {\nabla }}_{\mathbf {r} _{i}}\cdot {\boldsymbol {\nabla }}_{\mathbf {r} _{i}}={\frac {\partial ^{2}}{\partial x_{i}^{2}}}+{\frac {\partial ^{2}}{\partial y_{i}^{2}}}+{\frac {\partial ^{2}}{\partial z_{i}^{2}}}} . Since

11136-437: The electrons move in a quantum mechanical way. Within this framework the molecular Hamiltonian has been simplified to the so-called clamped nucleus Hamiltonian , also called electronic Hamiltonian , that acts only on functions of the electronic coordinates. Once the Schrödinger equation of the clamped nucleus Hamiltonian has been solved for a sufficient number of constellations of the nuclei, an appropriate eigenvalue (usually

11264-658: The end of 2023, has raised over $ 6.2 billion. Berkeley is a large, primarily residential research university with a majority of its enrolment in undergraduate programs but also offering a comprehensive doctoral program. The university has been accredited by the Western Association of Schools and Colleges Senior College and University Commission since 1949. The university operates on a semester calendar and awarded 8,725 bachelor's, 3,286 master's or professional and 1,272 doctoral degrees in 2018–2019. There are 1,789 full-time and 886 part-time faculty members among

11392-527: The equilibrium reciprocal moments of inertia on the diagonal. If all q s would be zero, only the kinetic energies of translation and rigid rotation would survive. The potential-like term U is the Watson term : U = − 1 8 ∑ α = 1 3 μ α α {\displaystyle U=-{\frac {1}{8}}\sum _{\alpha =1}^{3}\mu _{\alpha \alpha }} proportional to

11520-419: The first step of the Born–Oppenheimer approximation . See Ref. for a thorough discussion of the mathematical properties of the Coulomb Hamiltonian. Also it is discussed in this paper whether one can arrive a priori at the concept of a molecule (as a stable system of electrons and nuclei with a well-defined geometry) from the properties of the Coulomb Hamiltonian alone. The clamped nucleus Hamiltonian, which

11648-842: The history of the western part of North America, with an emphasis on California, Mexico and Central America. The Bancroft Library also houses the Mark Twain Papers, the Oral History Center, the Center for the Tebtunis Papyri, and the University Archives. In his memoirs, Clark Kerr records Berkeley's rise in the rankings (according to the National Academies ) during the 20th century. The school's first ranking in 1906 placed it among

11776-418: The hydrogen molecule. Almost all calculations of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first devised by Born and Oppenheimer . The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian and one considers the remaining Hamiltonian as a Hamiltonian of electrons only. The stationary nuclei enter the problem only as generators of an electric potential in which

11904-437: The internal (vibrational) motions, it is common to return at this point to classical mechanics and to formulate the classical kinetic energy corresponding to these motions of the nuclei. Classically it is easy to separate the translational—center of mass—motion from the other motions. However, the separation of the rotational from the vibrational motion is more difficult and is not completely possible. This ro-vibrational separation

12032-414: The introduction of a new degree of freedom for electrons and nuclei, namely spin . This empirical concept was given a theoretical basis by Paul Dirac when he introduced a relativistically correct ( Lorentz covariant ) form of the one-particle Schrödinger equation. The Dirac equation predicts that spin and spatial motion of a particle interact via spin–orbit coupling . In analogy spin-other-orbit coupling

12160-419: The invariance of V under rotation and translation that six of the eigenvectors of F (last six rows of Q ) have eigenvalue zero (are zero-frequency modes). They span the external space . The first 3 N − 6 rows of Q are—for molecules in their ground state—eigenvectors with non-zero eigenvalue; they are the internal coordinates and form an orthonormal basis for a (3 N  - 6)-dimensional subspace of

12288-467: The kinetic energy operator becomes, T = − ℏ 2 2 ∑ i = 1 N ∑ α = 1 3 ∂ 2 ∂ ρ i α 2 . {\displaystyle T=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}\sum _{\alpha =1}^{3}{\frac {\partial ^{2}}{\partial \rho _{i\alpha }^{2}}}.} If we make

12416-527: The kinetic energy operator is an inner product, it is invariant under rotation of the Cartesian frame with respect to which x i , y i , and z i are expressed. In the 1920s much spectroscopic evidence made it clear that the Coulomb Hamiltonian is missing certain terms. Especially for molecules containing heavier atoms, these terms, although much smaller than kinetic and Coulomb energies, are nonnegligible. These spectroscopic observations led to

12544-471: The lowest) can be seen as a function of the nuclear coordinates, which leads to a potential energy surface . In practical calculations the surface is usually fitted in terms of some analytic functions. In the second step of the Born–Oppenheimer approximation the part of the full Coulomb Hamiltonian that depends on the electrons is replaced by the potential energy surface. This converts the total molecular Hamiltonian into another Hamiltonian that acts only on

12672-637: The maintenance of the six Cal flags, the large California banner overhanging the Memorial Stadium Student Section and Haas Pavilion , the California Victory Cannon, Card Stunts and The Big "C" among other duties. The Rally Committee is also responsible for safekeeping of the Stanford Axe when it is in Cal's possession. Overlooking the main Berkeley campus from the foothills in the east, The Big "C"

12800-523: The medical schools at the University of California, San Francisco and Stanford University . The university offers Master of Arts , Master of Science , Master of Fine Arts , and PhD degrees in addition to professional degrees such as the Juris Doctor , Master of Business Administration , Master of Public Health , and Master of Design . The university awarded 963 doctoral degrees and 3,531 master's degrees in 2017. Admission to graduate programs

12928-559: The nation's first ROTC programs was established at Berkeley and its School of Military Aeronautics began training pilots, including Jimmy Doolittle . In 1926, future Fleet Admiral Chester W. Nimitz established the first Naval ROTC unit at Berkeley. Berkeley ROTC alumni include former Secretary of Defense Robert McNamara , Army Chief of Staff Frederick C. Weyand , sixteen other general officers , ten Navy flag officers , and AFROTC alumna Captain Theresa Claiborne . In

13056-426: The nation. Berkeley doctoral programs that received a #1 ranking included English, German, Political Science, Geography, Agricultural and Resource Economics, Mathematics, Physics, Chemistry, Biochemistry, Molecular Biology, Genetics, Genomics, Epidemiology, Plant Biology, Computer Science, Electrical Engineering, Mechanical Engineering, and Civil and Environmental Engineering. For Fall 2022, Berkeley's total enrollment

13184-759: The nuclear configuration space R , the internal space . The zero-frequency eigenvectors are orthogonal to the eigenvectors of non-zero frequency. It can be shown that these orthogonalities are in fact the Eckart conditions . The kinetic energy expressed in the internal coordinates is the internal (vibrational) kinetic energy. With the introduction of normal coordinates q t ≡ ∑ i = 1 N ∑ α = 1 3 Q t , i α ρ i α , {\displaystyle q_{t}\equiv \sum _{i=1}^{N}\sum _{\alpha =1}^{3}\;Q_{t,i\alpha }\rho _{i\alpha },}

13312-447: The nuclear coordinates. In the case of a breakdown of the Born–Oppenheimer approximation —which occurs when energies of different electronic states are close—the neighboring potential energy surfaces are needed, see this article for more details on this. The nuclear motion Schrödinger equation can be solved in a space-fixed (laboratory) frame , but then the translational and rotational (external) energies are not accounted for. Only

13440-596: The option of living in one of the twenty cooperative houses participating in the Berkeley Student Cooperative (BSC), a nonprofit housing cooperative network consisting of 20 residences and 1250 member-owners. About three percent of undergraduate men and nine percent of undergraduate women—or 3,400 of total undergraduates—are active in Berkeley's Greek system. University-sanctioned fraternities and sororities comprise over 60 houses affiliated with four Greek councils. The Associated Students of

13568-749: The original site of the Campus Arboretum. Trees in the area date from the founding of the university. The campus features numerous wooded areas, including: Founders' Rock , Faculty Glade, Grinnell Natural Area, and the Eucalyptus Grove, which is both the tallest stand of such trees in the world and the tallest stand of hardwood trees in North America. The campus sits on the Hayward Fault , which runs directly through California Memorial Stadium. The official university mascot

13696-424: The required quantum mechanical kinetic energy operator. When we apply this recipe to Cartesian coordinates, which have unit metric, the same kinetic energy is obtained as by application of the quantization rules . The nuclear motion Hamiltonian was obtained by Wilson and Howard in 1936, who followed this procedure, and further refined by Darling and Dennison in 1940. It remained the standard until 1968, when Watson

13824-419: The theory, particularly the relationship between the first- and second-order perturbative contributions. Perhaps the most significant application of the RKKY theory has been to the theory of giant magnetoresistance (GMR). GMR was discovered when the coupling between thin layers of magnetic materials separated by a non-magnetic spacer material was found to oscillate between ferromagnetic and antiferromagnetic as

13952-887: The third term. The term V 0 can be absorbed in the energy (gives a new zero of energy). The second term is vanishing because of the equilibrium condition. The remaining term contains the Hessian matrix F of V , which is symmetric and may be diagonalized with an orthogonal 3 N  × 3 N matrix with constant elements: Q F Q T = Φ with Φ = diag ⁡ ( f 1 , … , f 3 N − 6 , 0 , … , 0 ) . {\displaystyle \mathbf {Q} \mathbf {F} \mathbf {Q} ^{\mathrm {T} }={\boldsymbol {\Phi }}\quad {\text{with}}\quad {\boldsymbol {\Phi }}=\operatorname {diag} (f_{1},\dots ,f_{3N-6},0,\ldots ,0).} It can be shown from

14080-435: The top six schools ("Big Six") in the nation. In 1934, it ranked second, tied with Columbia and the University of Chicago , behind only Harvard ; in 1957, it was ranked as the only school second to Harvard. In 1964, Berkeley was named the "best balanced distinguished university", meaning the school had not only the most top departments but also the highest percentage of top ranking departments in its school. The school in 1993

14208-809: The trace of the effective reciprocal inertia tensor. The fourth term in the Watson Hamiltonian is the kinetic energy associated with the vibrations of the atoms (nuclei) expressed in normal coordinates q s , which as stated above, are given in terms of nuclear displacements ρ iα by q s = ∑ i = 1 N ∑ α = 1 3 Q s , i α ρ i α for s = 1 , … , 3 N − 6. {\displaystyle q_{s}=\sum _{i=1}^{N}\sum _{\alpha =1}^{3}Q_{s,i\alpha }\rho _{i\alpha }\quad {\text{for}}\quad s=1,\ldots ,3N-6.} Finally V

14336-524: The two atoms' spins. Tadao Kasuya from Nagoya University later proposed that a similar indirect exchange coupling could occur with localized inner d-electron spins instead of nuclei. This theory was expanded more completely by Kei Yosida of the UC Berkeley, to give a Hamiltonian that describes (d-electron spin)–(d-electron spin), (nuclear spin)–(nuclear spin), and (d-electron spin)–(nuclear spin) interactions. J.H. Van Vleck clarified some subtleties of

14464-541: The university also owns land to the northwest of the main campus, a married student housing complex in the nearby town of Albany ("Albany Village" and the "Gill Tract"), and a field research station several miles to the north in Richmond, California . The campus is home to several museums including the University of California Museum of Paleontology , the Berkeley Art Museum and Pacific Film Archive , and

14592-538: The university is also a producer of Rhodes Scholars , Marshall Scholars , and Fulbright Scholars . Made possible by President Lincoln 's signing of the Morrill Act in 1862, the University of California was founded in 1868 as the state's first land-grant university, inheriting the land and facilities of the private College of California and the federal-funding eligibility of a public agricultural, mining, and mechanical arts college. The Organic Act states that

14720-626: The university since 1891, performs at every home football game and at select road games as well. A smaller subset of the Cal Band, the Straw Hat Band, performs at basketball games, volleyball games, and other campus and community events. The UC Rally Committee, formed in 1901, is the official guardian of California's Spirit and Traditions. Wearing their traditional blue and gold rugbies, Rally Committee members can be seen at all major sporting and spirit events. Committee members are charged with

14848-421: The university's academic enterprise which is organized into fifteen colleges and schools that comprise 180 departments and 80 interdisciplinary units offering over 350 degree programs. Colleges serve both undergraduate and graduate students, while schools are generally graduate only, though some offer undergraduate majors or minors: The four-year, full-time undergraduate program offers 107 bachelor's degrees across

14976-561: The upscale residential neighborhoods of the Berkeley Hills . Immediately southeast of campus lies fraternity row and beyond that the Clark Kerr Campus and an upscale residential area named Claremont . The area south of the university includes student housing and Telegraph Avenue , one of Berkeley's main shopping districts with stores, street vendors and restaurants catering to college students and tourists. In addition,

15104-671: The vibrational (internal) part of the Hamiltonian for the nuclear motion becomes in the harmonic approximation H ^ nuc ≈ 1 2 ∑ t = 1 3 N − 6 [ − ℏ 2 ∂ 2 ∂ q t 2 + f t q t 2 ] . {\displaystyle {\hat {H}}_{\text{nuc}}\approx {\frac {1}{2}}\sum _{t=1}^{3N-6}\left[-\hbar ^{2}{\frac {\partial ^{2}}{\partial q_{t}^{2}}}+f_{t}q_{t}^{2}\right].} The corresponding Schrödinger equation

15232-455: The west of the central campus is the downtown business district of Berkeley ; to the northwest is the neighborhood of North Berkeley, including the so-called Gourmet Ghetto , a commercial district known for high quality dining due to the presence of such world-renowned restaurants as Chez Panisse . Immediately to the north is a quiet residential neighborhood known as Northside with a large graduate student population; situated north of that are

15360-486: Was 45,745: 32,831 undergraduate and 12,914 graduate students, with women accounting for 56% of undergraduates and 49% of graduate and professional students. It had 128,226 freshman applicants and accepted 14,614 (11.4%). Among enrolled freshman, the average unweighted GPA was 3.90. Berkeley's enrollment of National Merit Scholars was third in the nation until 2002, when participation in the National Merit program

15488-682: Was Frenchman Émile Bénard , who refused to personally supervise the implementation of his plan and the task was subsequently given to architecture professor John Galen Howard . Howard designed over twenty buildings, which set the tone for the campus up until its expansion in the 1950s and 1960s. The structures forming the "classical core" of the campus were built in the Beaux-Arts Classical style, and include Hearst Greek Theatre , Hearst Memorial Mining Building , Doe Memorial Library , California Hall, Wheeler Hall , Le Conte Hall, Gilman Hall, Haviland Hall, Wellman Hall, Sather Gate , and

15616-553: Was able to simplify it drastically by commuting through the derivatives the determinant of the metric tensor. We will give the ro-vibrational Hamiltonian obtained by Watson, which often is referred to as the Watson Hamiltonian . Before we do this we must mention that a derivation of this Hamiltonian is also possible by starting from the Laplace operator in Cartesian form, application of coordinate transformations, and use of

15744-606: Was discontinued. For 2019, Berkeley ranked fourth in enrollment of recipients of the National Merit $ 2,500 Scholarship (132 scholars). 27% of admitted students receive federal Pell grants . Berkeley students are eligible for a variety of public and private financial aid. Inquiries are processed through the Financial Aid and Scholarships Office, although schools such as the Haas School of Business and Berkeley Law , have their own financial aid offices. Much of

15872-447: Was first achieved by Eckart in 1935 by imposing by what is now known as Eckart conditions . Since the problem is described in a frame (an "Eckart" frame) that rotates with the molecule, and hence is a non-inertial frame , energies associated with the fictitious forces : centrifugal and Coriolis force appear in the kinetic energy. In general, the classical kinetic energy T defines the metric tensor g = ( g ij ) associated with

16000-493: Was formed in 2005 and broadcasts online. It is run by students with a variety of backgrounds and majors. Since the mid-2010s, it has been a program of the ASUC . Berkeley's independent student-run newspaper is The Daily Californian . Founded in 1871, The Daily Cal became independent in 1971 after the campus administration fired three senior editors for encouraging readers to take back People's Park . The Daily Californian has both

16128-572: Was in 1874, admitted in the first class to include women in 1870. Beginning in 1891, Phoebe Apperson Hearst funded several programs and new buildings and, in 1898, sponsored an international competition in Antwerp , where French architect Émile Bénard submitted the winning design for a campus master plan. Although the University of California system does not have an official flagship campus , many scholars and experts consider Berkeley to be its unofficial flagship. It shares this unofficial status with

16256-484: Was introduced. The fact that particle spin has some of the characteristics of a magnetic dipole led to spin–spin coupling . Further terms without a classical counterpart are the Fermi-contact term (interaction of electronic density on a finite size nucleus with the nucleus), and nuclear quadrupole coupling (interaction of a nuclear quadrupole with the gradient of an electric field due to the electrons). Finally

16384-549: Was the only remaining member of the original 1906 "Big Six", along with Harvard; in that year Berkeley ranked first. The American Council on Education , a private non-profit association, ranked Berkeley tenth in 1934. However, by 1942, private funding had helped Berkeley rise to second place, behind only Harvard, based on the number of distinguished departments. In 1985, Yale University admissions officer Richard Moll published Public Ivies: A Guide to America's Best Public Undergraduate Colleges and Universities which named Berkeley

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