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Density Functional

Density Functional Theory (DFT) is a computational quantum mechanical modeling method used to investigate the electronic structure of many-body systems, particularly atoms, molecules, and solids. The core idea of DFT is that the properties of a system can be determined by its electron density rather than its wave function. This allows for significant simplifications in calculations, as the electron density ρ(r)\rho(\mathbf{r})ρ(r) is a function of three spatial variables, while a wave function depends on the number of electrons and can be much more complex.

DFT employs functionals, which are mathematical entities that map functions to real numbers, to express the energy of a system in terms of its electron density. The total energy E[ρ]E[\rho]E[ρ] can be expressed as:

E[ρ]=T[ρ]+V[ρ]+Exc[ρ]E[\rho] = T[\rho] + V[\rho] + E_{xc}[\rho]E[ρ]=T[ρ]+V[ρ]+Exc​[ρ]

Here, T[ρ]T[\rho]T[ρ] is the kinetic energy functional, V[ρ]V[\rho]V[ρ] is the classical electrostatic interaction energy, and Exc[ρ]E_{xc}[\rho]Exc​[ρ] represents the exchange-correlation energy, capturing all quantum mechanical interactions. DFT's ability to provide accurate predictions for the properties of materials while being computationally efficient makes it a vital tool in fields such as chemistry, physics, and materials science.

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Lebesgue-Stieltjes Integral

The Lebesgue-Stieltjes integral is a generalization of the Lebesgue integral, which allows for integration with respect to a more general type of measure. Specifically, it integrates a function fff with respect to another function ggg, where ggg is a non-decreasing function. The integral is denoted as:

∫abf(x) dg(x)\int_a^b f(x) \, dg(x)∫ab​f(x)dg(x)

This formulation enables the integration of functions that may not be absolutely continuous, thereby expanding the types of functions and measures that can be integrated. It is particularly useful in probability theory and in the study of stochastic processes, as it allows for the integration of random variables with respect to cumulative distribution functions. The properties of the integral, including linearity and monotonicity, make it a powerful tool in analysis and applied mathematics.

Single-Cell Rna Sequencing

Single-Cell RNA Sequencing (scRNA-seq) is a groundbreaking technique that enables the analysis of gene expression at the individual cell level. Unlike traditional RNA sequencing, which averages the gene expression across a population of cells, scRNA-seq allows researchers to capture the unique transcriptomic profile of each cell. This is particularly important for understanding cellular heterogeneity in complex tissues, discovering rare cell types, and investigating cellular responses to various stimuli.

The process typically involves isolating single cells from a sample, converting their RNA into complementary DNA (cDNA), and then sequencing this cDNA to quantify the expression levels of genes. The resulting data can be analyzed using various bioinformatics tools to identify distinct cell populations, infer cellular states, and map developmental trajectories. Overall, scRNA-seq has revolutionized our approach to studying cellular function and diversity in health and disease.

Minimax Theorem In Ai

The Minimax Theorem is a fundamental principle in game theory and artificial intelligence, particularly in the context of two-player zero-sum games. It states that in a zero-sum game, where one player's gain is equivalent to the other player's loss, there exists a strategy that minimizes the possible loss for a worst-case scenario. This can be expressed mathematically as follows:

minimax(A)=max⁡s∈Smin⁡a∈AV(s,a)\text{minimax}(A) = \max_{s \in S} \min_{a \in A} V(s, a)minimax(A)=s∈Smax​a∈Amin​V(s,a)

Here, AAA represents the set of strategies available to Player A, SSS represents the strategies available to Player B, and V(s,a)V(s, a)V(s,a) is the payoff function that details the outcome based on the strategies chosen by both players. The theorem is particularly useful in AI for developing optimal strategies in games like chess or tic-tac-toe, where an AI can evaluate the potential outcomes of each move and choose the one that maximizes its minimum gain while minimizing its opponent's maximum gain, thus ensuring the best possible outcome under uncertainty.

Caratheodory Criterion

The Caratheodory Criterion is a fundamental theorem in the field of convex analysis, particularly used to determine whether a set is convex. According to this criterion, a point xxx in Rn\mathbb{R}^nRn belongs to the convex hull of a set AAA if and only if it can be expressed as a convex combination of points from AAA. In formal terms, this means that there exists a finite set of points a1,a2,…,ak∈Aa_1, a_2, \ldots, a_k \in Aa1​,a2​,…,ak​∈A and non-negative coefficients λ1,λ2,…,λk\lambda_1, \lambda_2, \ldots, \lambda_kλ1​,λ2​,…,λk​ such that:

x=∑i=1kλiaiand∑i=1kλi=1.x = \sum_{i=1}^{k} \lambda_i a_i \quad \text{and} \quad \sum_{i=1}^{k} \lambda_i = 1.x=i=1∑k​λi​ai​andi=1∑k​λi​=1.

This criterion is essential because it provides a method to verify the convexity of a set by checking if any point can be represented as a weighted average of other points in the set. Thus, it plays a crucial role in optimization problems where convexity assures the presence of a unique global optimum.

Charge Transport In Semiconductors

Charge transport in semiconductors refers to the movement of charge carriers, primarily electrons and holes, within the semiconductor material. This process is essential for the functioning of various electronic devices, such as diodes and transistors. In semiconductors, charge carriers are generated through thermal excitation or doping, where impurities are introduced to create an excess of either electrons (n-type) or holes (p-type). The mobility of these carriers, which is influenced by factors like temperature and material quality, determines how quickly they can move through the lattice. The relationship between current density JJJ, electric field EEE, and carrier concentration nnn is described by the equation:

J=q(nμnE+pμpE)J = q(n \mu_n E + p \mu_p E)J=q(nμn​E+pμp​E)

where qqq is the charge of an electron, μn\mu_nμn​ is the mobility of electrons, and μp\mu_pμp​ is the mobility of holes. Understanding charge transport is crucial for optimizing semiconductor performance in electronic applications.

Is-Lm Model

The IS-LM model is a fundamental tool in macroeconomics that illustrates the relationship between interest rates and real output in the goods and money markets. The model consists of two curves: the IS curve, which represents the equilibrium in the goods market where investment equals savings, and the LM curve, which represents the equilibrium in the money market where money supply equals money demand.

The intersection of the IS and LM curves determines the equilibrium levels of interest rates and output (GDP). The IS curve is downward sloping, indicating that lower interest rates stimulate higher investment and consumption, leading to increased output. In contrast, the LM curve is upward sloping, reflecting that higher income levels increase the demand for money, which in turn raises interest rates. This model helps economists analyze the effects of fiscal and monetary policies on the economy, making it a crucial framework for understanding macroeconomic fluctuations.