Slater's determinant

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The Slater determinant is a mathematical technique of quantum mechanics that is used to generate antisymmetric wave functions that describe the collective states of several fermions and that satisfy the Pauli exclusion principle.

This type of determinants take their name from John C. Slater, a U.S. physicist and theoretical chemist who proposed their use in order to ensure that the electronic wave function is antisymmetrical regarding the exchange of two electrons. Slater's determinants are built from monoelectronic wave functions called spinal-orbital ${displaystyle chi (mathbf {x})}$ Where ${mathbf {x}}$ represents the coordinates of position and sphin of the electron. As a consequence of the properties of the determinants, two electrons cannot be described by the same spin-orbital as it would mean that the wave function is overturned throughout the space.

Two particles

In order to illustrate its functioning we can consider the simplest case of two particles. Yeah. ${displaystyle {boldsymbol {x}}_{1}}$ and ${displaystyle {boldsymbol {x}}_{2}}$ are the coordinates (spatial and spinal) of particle 1 and particle 2 respectively, can generate the function of collective wave $Psi$ as the product of the individual wave functions of each particle, i.e.

${displaystyle Psi ({boldsymbol {x}}_{1},{boldsymbol {x}}_{2})=chi _{1}({boldsymbol {x}}_{1})chi _{2}({boldsymbol {x}}_{2}).}$

This expression is called the Hartree product, and it is the simplest wave function that we can write within the orbital approximation. In fact, this type of wavefunction is not valid for the representation of collective states of fermions since this wavefunction is not antisymmetric to a particle exchange. The function must satisfy the following condition

${displaystyle Psi ({boldsymbol {x}}_{1},{boldsymbol {x}}_{2})=-Psi ({boldsymbol {x}}_{2},{boldsymbol {x}}_{1}).}$

It is easy to check that although the above Hartree product is not antisymmetric with respect to particle exchange, the following linear combination of these products is

${displaystyle Psi ({boldsymbol {x}}_{1},{boldsymbol {x}}_{2})={frac {1}{sqrt {2}}}left[chi _{1}({boldsymbol {x}}_{1})chi _{2}({boldsymbol {x}}_{2})-chi _{1}({boldsymbol {x}}_{2})chi _{2}({boldsymbol {x}}_{1})right],}$

where we have included a factor so that the wave function is conveniently normalized. This last equation can be rewritten as a determinant as follows

${displaystyle Psi ({boldsymbol {x}}_{1},{boldsymbol {x}}_{2})={frac {1}{sqrt {2}}}left|{begin{matrix}chi _{1}({boldsymbol {x}}_{1})&chi _{1}({boldsymbol {x}}_{2})\chi _{2}({boldsymbol {x}}_{1})&chi _{2}({boldsymbol {x}}_{2})end{matrix}}right|,}$

known as the determinant of Slater functions ${displaystyle chi _{1}}$ and ${displaystyle chi _{2}}$ . Therefore this wave function in addition to being antisymmetric, considers that the two electrons are indisputable particles. The generated functions thus have the property of annulment if two of the wave functions of a particle are equal or, what is equivalent, two of the femiions are described by the same orbital spine. This is equivalent to meeting Pauli's exclusion principle.

N particles

To define the Slater determinant of n fermions, it is convenient to previously define the Hartree Product (ph) of n spin-orbitals, which is defined as the following product

${displaystyle |chi _{i_{1}}(x_{1}),chi _{i_{2}}(x_{2}),dotschi _{i_{n}}(x_{n})rangle _{ph}=|chi _{i_{1}}(x_{1})rangle otimes |chi _{i_{2}}(x_{2})otimes dots otimes |chi _{i_{n}}(x_{n})rangle }$

Also defining the antisymmetrizer as

${displaystyle A={frac {1}{sqrt {n!}}}sum _{alpha }(-1)^{alpha }P_{alpha }}$

where the summary runs on all the permutations ${displaystyle P_{alpha }}$ possible. The factor ${displaystyle (-1)^{alpha }}$ incorporates a positive sign if the permutation is pair and a negative sign if it is odd.

As an example, to understand how the permutation operator works we can consider the case of 3 particles

${displaystyle P_{(3,1,2)}|chi _{i_{1}}(x_{1}),chi _{i_{2}}(x_{2}),chi _{i_{3}}(x_{3})rangle _{ph}=|chi _{i_{3}}(x_{1}),chi _{i_{1}}(x_{2}),chi _{i_{2}}(x_{3})rangle _{ph}}$

Having defined the anti-symmeterizer $A$ then the determinant of Slater (DS) is written according to the expression:

${displaystyle |chi _{i_{1}},chi _{i_{2}},dotschi _{i_{n}}rangle _{DS}=A|chi _{i_{1}}(x_{1}),chi _{i_{2}}(x_{2}),dotschi _{i_{n}}(x_{n})rangle _{ph}}$

Note that the square root applied to the factor n! in the definition of the antisymmetrizer given previously allows the Slater determinant to be automatically normalized.

Some properties of the antisymmetrizer

Remembering the definition

${displaystyle A={frac {1}{sqrt {n!}}}sum _{alpha }(-1)^{alpha }P_{alpha }}$

this operator is Hermitian and satisfies the relation

${displaystyle A^{2}={sqrt {n!}}A}$

In some texts the following alternative definition is used for the antisymmetrizer

${displaystyle A'={frac {1}{n!}}sum _{alpha }(-1)^{alpha }P_{alpha }}$

which has the advantage of simplifying the previous expression, obtaining

${displaystyle A'^{2}=A'}$

but the disadvantage of complicating a bit the expression that relates the Hartree product and the Slater determinant

${displaystyle |chi _{i_{1}},chi _{i_{2}},dotschi _{i_{n}}rangle _{DS}={sqrt {n!}}A'|chi _{i_{1}}(x_{1}),chi _{i_{2}}(x_{2}),dotschi _{i_{n}}(x_{n})rangle _{ph}}$

Slater determinants and their relationship with the Hartree-Fock method

In the Hartree-Fock method, a single Slater determinant is used as an approximation to the electronic wave function, similar methods are called monodeterminant. In more precise calculation methods, such as configuration interaction or MCSCF, linear superpositions of Slater determinants are used, and are called multideterminant methods.

Matrix elements between Slater determinants

Given a Slater determinant consisting of n orthonormal spin-orbitals:

${displaystyle |Psi rangle =|chi _{a},chi _{b},dotschi _{f}rangle _{DS}=|a,b,dotsfrangle _{DS}}$

(where the ellipsis represents n-3 distinct orbital spins)

a Slater determinant can be defined that differs in a spin-orbital from the previous one:

${displaystyle |Psi _{a}^{a'}rangle =|chi _{a'},chi _{b},dotschi _{f}rangle _{DS}=|a',b,dotsfrangle _{DS}}$

Similarly, one can define a Slater determinant that differs by 2 orbital spins from the first one:

${displaystyle |Psi _{ab}^{a'b'}rangle =|chi _{a'},chi _{b'},dotschi _{f}rangle _{DS}=|a',b',dotsfrangle _{DS}}$

Operators of a body

The operators of a field are operators that act in the space of n particles that are constituted by a sum of operators that act in the spaces of a paticle for each of the coordinates:

${displaystyle {hat {O}}_{1}=sum _{i=1}^{n}{hat {h}}(i)}$

Example: any component of the total spatial angular momentum:

${displaystyle {hat {L}}_{z}=sum _{i=1}^{n}{hat {l}}_{z}(i)}$

similarly for any component of spin angular momentum.

Array elements

Expressions of the matrix elements in each case:

${displaystyle langle Psi |{hat {O}}_{1}|Psi rangle =sum _{min Psi }langle m|{hat {h}}|mrangle }$

${displaystyle langle Psi _{a}^{a'}|{hat {O}}_{1}|Psi rangle =langle a'|{hat {h}}|arangle }$

${displaystyle langle Psi _{ab}^{a'b'}|{hat {O}}_{1}|Psi rangle =0}$

For any one-field operator, when the Slater determinants differ by 2 or more spin-orbitals from each other, the matrix element between them is zero.

Two-body operators

The operators of two bodies (which act in the space of n particles), are constituted by a sum over the different pairs of particles and act in the respective spaces of two particles for each pair of coordinates:

$<math alttext="{displaystyle {hat {O}}_{2}=sum _{i=1}^{n}sum _{jO^ ^ 2=␡ ␡ i=1n␡ ␡ j.ing^ ^ (i,j){displaystyle {hat {O}}_{2}=sum _{i=1}^{n}sum _{jι}{n}{hat {g}}(i,j)}}<img alt="{displaystyle {hat {O}}_{2}=sum _{i=1}^{n}sum _{j$

Example: electron-electron interaction:

$<math alttext="{displaystyle {hat {V}}_{ee}={frac {e^{2}}{4pi epsilon _{0}}}sum _{i=1}^{n}sum _{jV^ ^ ee=e24π π ε ε 0␡ ␡ i=1n␡ ␡ j.in1日本語r! ! i− − r! ! j日本語{displaystyle {hat {V}}_{ee}={frac {e^{2}{4pi epsilon _{0}}}}{i=1}{n}{n}sum _{j bound}{n}{n}{frac {1}{1}{frac}{bar {r}}}}}}{{ }}}}{{{{ {s}}}}}}}}{ { { { { { { { }}}{ { { }}}}{ { { { { { }}}}}{ { { { { { { { }}}}}}{ { { }}{ { { { { }}}}}}{ }{ { { { { { { { { { }{ }}}{ { }{ }{ }{ }}}}}}}{<img alt="{displaystyle {hat {V}}_{ee}={frac {e^{2}}{4pi epsilon _{0}}}sum _{i=1}^{n}sum _{j$

Array elements

Expressions of the matrix elements in each case:

$<math alttext="{displaystyle langle Psi |{hat {O}}_{2}|Psi rangle =sum _{min Psi }sum _{(p 日本語O^ ^ 2日本語 =␡ ␡ m한 한 ␡ ␡ (p.m)한 한 ( m,p日本語phg^ ^ (1,2)日本語m,p ph− − m,p日本語phg^ ^ (1,2)日本語p,m ph){displaystyle langle Psi 日本語{hat {O}}_{2}{Psi rangle =sum _{min Psi }sum _{(pparentm)in Psi}{langle m,p gross_{ph}{ph}{,{hat {g}{1}{cH00FFFFFFFFFFFF00}{cH00FFFFFF00}}{cH00}{cH00}}{cH00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}}{cH00}}{cH00}{cH00}{cH00FFFFFFFFFFFFFFFFFF00}}{cH00}{cH00}{cH00FFFFFFFFFFFFFFFFFF00}{cH00FFFFFF00}{cH00}{cH00FFFFFFFFFFFFFFFFFFFFFF00}{cH<img alt="{displaystyle langle Psi |{hat {O}}_{2}|Psi rangle =sum _{min Psi }sum _{(p$

${displaystyle langle Psi _{a}^{a'}|{hat {O}}_{2}|Psi rangle =sum _{min Psi }left(langle m,a'|_{ph},{hat {g}}(1,2)|m,arangle _{ph}-langle m,a'|_{ph},{hat {g}}(1,2)|a,mrangle _{ph}right)}$

${displaystyle langle Psi _{a,b}^{a'b'}|{hat {O}}_{2}|Psi rangle =langle a',b'|_{ph},{hat {g}}(1,2)|a,brangle _{ph}-langle a',b'|_{ph},{hat {g}}(1,2)|b,arangle _{ph}}$

${displaystyle langle Psi _{a,b,c}^{a'b'c'}|{hat {O}}_{2}|Psi rangle =0}$

For any two-body operator, when the Slater determinants differ by 3 or more spin-orbitals from each other, the matrix element between them is zero.

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